Electron optics with dirac fermions: electron transport in

monolayer and bilayer graphene through magnetic barrier and

their superlattices

Neetu Agrawal (Garg)1, Sankalpa Ghosh2 and Manish Sharma1

1Centre for Applied Research in Electronics,

Indian Institute of Technology Delhi, New Delhi-110016, India and

2Department of Physics, Indian Institute of Technology Delhi, New Delhi-110016, India∗

(Dated: April 6, 2018)

Abstract

In this review article we discuss the recent progress in studying ballistic transport for charge

carriers in graphene through highly inhomogenous magnetic field known as magnetic barrier in

combination with gate voltage induced electrostatic potential. Starting with cases for a single or

double magnetic barrier we also review the progress in understanding electron transport through

the superlattices created out of such electromagnetic potential barriers and discuss the possibility

of experimental realization of such systems. The emphasis is particularly on the analogy of such

transport with propagation of light wave through medium with alternating dielectric constant.

In that direction we discuss electron analogue of optical phenomena like fabry perot resonances,

negative refraction, Goos-Hanchen effect, beam collimation in such systems and explain how such

analogy is going to be useful for device generation. The resulting modification of band structure of

dirac fermions, the emergence of additional dirac points was also discussed accompanied by brief

section on the interconvertibility of electric and magnetic field for relativistic dirac fermions. We

also discuss the effect of such electromagnetic potential barrier on bilayer graphene in a similar

framework.

PACS numbers: 81.05. Ue, 73.63.-b, 78.20.Ci, 42.25.Gy, 73.43.Qt

∗Electronic address: sankalpa@physics.iitd.ac.in

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Contents

I. Introduction 3

A. Low energy description of charge carriers in graphene 6

II. Electron transport in graphene : Optical analogy 9

A. Electron optics in the presence of scalar potential barriers 9

B. Klein Tunneling 11

III. Electron transport in the presence of inhomogenous magnetic field

profile 14

A. Producing inhomogenous magnetic field profile: Experimental strategies 15

B. Magnetic confinement of massless dirac fermions in graphene 18

IV. Massless dirac fermions in inhomogenous magnetic fields: An analogy

with light propagation 20

A. Magnetic vector potential (MVP) barrier 20

1. Electron optics with MVP barrier 21

2. MVP barrier: Transmission probability 23

B. Combined MVP barrier and electrostatic potential (EMVP) barrier 25

1. Electron optics with EMVP barrier 27

2. EMVP barrier: Transmission probability 32

3. Conductance: Effect of various EMVP Barriers on Transport 34

4. Generalisation to MVP/EMVP barriers of any arbitrary shape 35

C. Quantum Goos-Hanchen Shift in single MVP and EMVP barriers 35

1. GH shift at p-n interface 38

2. GH shift for MVP/EMVP barriers 39

3. Lateral shift in the transmitted wave 41

D. Fabry-Perot Interference in Graphene Heterojunctions: Effect of magnetic field 43

V. Periodic lattice of MVP barriers 47

A. Bandstructure modification 47

1. Dispersion in the vicinity of neutrality point 51

2. finite energy Dirac points 52

2

VI. The transformation of Magnetic to Electric Field for dirac fermions in

graphene: a case for relativistic invariance 54

VII. Magnetic barriers in bilayer graphene 56

A. Transmission through magnetic barriers 56

B. Conductance 61

C. Generalisation beyond bilayer 64

VIII. Conclusion 67

IX. Acknowledgements 69

References 69

I. INTRODUCTION

Physically Maxwell’s equation that describes the propagation of electromagnetic

wave/light through free space and dielectric medium, namely

∂2(E,B)

∂t2=

1

µε∇2(E,B) (1)

and Schrodinger equation that describes the time evolution of probability amplitude or de

Broglie wave in quantum systems, namely

i~∂Ψ

∂t= − ~2

2m∇2Ψ + VΨ (2)

correspond to the different type of phenomena. However, their striking mathematical sim-

ilarity indicates that a large number of wave like phenomena will occur in either of these

cases. Whereas the motion of a isolated single electron in presence of potential barrier is

described by the wave Eq.(2), in real materials due to the scattering by other electrons,

impurities etc. the motion of the charge carriers is generally diffusive. Therefore, such wave

propagation based description of the electron transport becomes only meaningful if the elec-

tron mean free path is of the order the typical sample size. Such a transport regime is called

the ballistic transport regime and in that regime the similarity between the propagations of

transport electrons and electromagnetic wave or light promises rich dividend [1, 2].

3

A particular consequence of the similarity between the wave equations (1) and (2) is

that for a monochromatic wave with a given frequency in Eq. 1 the role played by the

dielectric constant of a given medium has its corresponding analogue in the the potential

landscape for a stationary solution of the Schrodinger electron described by Eq. 2. Thus

some of the effects that one obtains by spatially modulating the potential in a Schrodinger

equation can be reproduced for the light also by spatially modulating the dielectric constant

of the medium. Indeed this was pointed out in seventies in the pioneering work by Yariv and

collaborators [3] that the propagation of light through a medium with periodically modulated

dielectric constant will lead to the similar band structure of transport electron as observed

in Kronig-Penny model [4]. In this context it is particularly useful to point out that for

conventional two dimensional electron gas (2DEG), the analogy between transmission of de

Broglie waves satisfying the Schrodinger equation through a one-dimensional electrostatic

potential [5, 6] and light propagation in linear dilelectric medium is well established and led

to the development of a number of applications.

The recent discovery of graphene [7–10] added a new twist to this well established optical

analogy of ballistic electron transport and vice versa. In graphene, electrons near the Fermi

level, namely the transport electrons no more obeys quadratic dispersion law, a typical char-

acteristics of their non-relativistic nature, but rather obeys a linear dispersion relation at

or near the Fermi surface, an archetypical of the ultra relativistic massless particles. Hence

they are called massless Dirac fermions. It has now been established that transport of such

massless Dirac fermions in the presence of an electrostatic potential barrier is analogous to

negative refraction through metamaterials [11–13]. The relativistic behaviour of graphene

electrons also leads to Klein tunnelling [14], where a massless relativistic particle can tun-

nel through any potential barrier above the Fermi surface, invalidating the possibility of

confining it using such potential barriers.

In the context of electronic transport in graphene devices that are proposed for graphene

based electronic in future, the local carrier concentration is controlled by one or more local

gates. Particularly, in the cleanest devices which satisfy the criterion of ballistic transport,

transport signatures of the relativistic nature of the charge carriers of graphene are observed.

A vast body of theoretical work ( for a review see [15]) has already been devoted to the study

of multigated graphene superlattices which can form a basis for a new kind of electronic

optics based on graphene as an electronic metamaterial. However the inability of confining

4

the electrons using a electrostatic potential barrier severely limits the applicability of such

devices. A solution to that problem was proposed by De Martino et al. [16] when it

was pointed out that well localized magnetic field dubbed as magnetic barrier can confine

massless dirac fermions in graphene. This suggests that one way of making high mobility

graphene based electronic devices is to make locally gated structures in conjunction with

such magnetic barriers. Given this context it is extremely important to see what will happen

to the above mentioned optical analogy when it applies to the cases of such magnetic barriers.

However it may be pointed out that the study of transport properties of two dimensional

non relativistic electron system in the presence of a transverse magnetic field remains mostly

confined to the case where the magnetic field is uniform on the scale of sample size through

phenomena such as Integer and Fractional Quantum Hall effect. Study of electron transport

in inhomogenous magnetic field is relatively lesser known field [17], even though inhomoge-

nous magnetic field is being used for many other purpose for a long time, such as one in

the famous Stern Gerlach experiment [18]. In a homogenous magnetic field, the electrons

execute cyclotron motion, the direction of the wavevector continuously changes, hence ren-

dering any analogy with the monochromatic light propagation in dielectric medium is very

unlikely. In quantum mechanical language the electronic wavefunction is localized over the

scale of the magnetic length. Thus a direct analogy with a propagating wave is not possible.

This situation however changed, when it was pointed out [19] that an optical analogy

can be constructed for electron transport in graphene in presence of highly inhomogenous

magnetic field [20] where one does not get bound state solutions, but rather scattering state

solutions. This suggests that inhomogenous magnetic field not only confine massless dirac

fermions [16], but the ballistic transport through such barriers can also be understood in

terms of suitable optical analogy. Ref. [19] also showed that the analogy with light propa-

gation in medium with periodically modulated dielectric constant can again be constructed

by considering the transport of massless dirac fermions in graphene through the periodic

arrangement of such highly inhomogenous magnetic field. Subsequent work [21] showed that

a very rich transport regime can be realized for such graphene electrons by considering struc-

tures where a local gate voltage is used simultaneously with such magnetic barrier. When

such configurations are generalized to a superlattice structure, that significantly modifies

the band structure of graphene electrons. Subsequently a large volume of theoretical work

pointed out the relevance of such optical analogy and proposed device structures based on

5

that. Other properties such as Goos Hanchen shift of graphene electrons by these type of

barriers, collimation of the electron beam, generation of additional Dirac points was also

discussed in further works. It was also extended in the case of bilayer graphene. In the

current review article we plan to review these developments. The purpose is two fold. One,

is to review the general theoretical framework in a pedagogical way, highlighting a number

of significant work done in this subfield of graphene research. The other purpose is to indi-

cate in very clear terms the possibility of designing interesting experiment and novel device

structure that holds rich promises for graphene based electronics.

The organization of the article is as follows. After giving a brief review of low energy

description of charge carriers in graphene, first we explain how electron transport through

scalar electrostatic barriers and magnetic barriers is explained in the language of geometrical

optics. Next, we explain how such transport gets modified when various types of potential

barriers are created atop magnetic barriers. We discuss Goos-Hanchen shift and fabry perot

resonances in this type of structure. Then we provide an analysis of transport through

an infinite series of such barriers pointing towards an effective way of changing the band-

structure. Here we also discuss the collimation of the electron beam, emergence of extra

dirac point etc. We then briefly discuss the issue of interconvertibilty of electrostatic poten-

tial and vector potential due to magnetic field citing relativistic invariance of the equation

obeyed by transport electrons in graphene. Then we come to the issue of electron transport

through graphene bilayer in presence of such potential barriers. The dispersion relation for

graphene bilayer is very different from that of monolayer graphene. The charge carriers in

bilayer graphene have a parabolic energy spectrum, which means they are massive, similar

to conventional non relativistic electrons. On the other hand, due to the crystal structure

of graphene consisting of two sublattices, these particles are described by spinor wave func-

tions similar to that of monolayer graphene. We study how the transport of such electrons

in graphene bilayer is affected, again when exposed to combination of magnetic barriers and

voltages.

A. Low energy description of charge carriers in graphene

The interesting band structure of graphene was pointed out by Phil Wallace [7] way back

in 1947. After its experimental discovery by Geim, Novoselov, Kim and their collaborators

6

there are many excellent review articles (for example see [22] ) where the band structure,

the resulting dirac fermion nature of the charge carriers are detailed. Here we provide a

brief introduction to the low energy description of these charge carriers for completeness.

Graphene is a one atom thick planar sheet of sp2 bonded carbon atoms. The carbon atoms

are distributed at the edges of regular hexagons forming a honeycomb lattice. However, the

honeycomb lattice is not a Bravais lattice and, from a crystallographic point of view, has to

be described by two inter penetrating triangular lattices with two atoms (sublattice A and

B) per unit cell.

Each carbon atom has four valence electrons, three of which hybridize to form sp2 hy-

bridized orbitals to form sigma bonds with neighbouring atoms. However, the fourth un-

bound electron lies in the 2pz orbital which extends vertically above and below the plane.

It is this electron which interacts with the periodic field of the hexagonal crystal lattice of

graphene. If we denote the creation operator by a†i (b†i ) for an atom on the A(B) sublattice,

then the nearest neighbour tightbinding Hamiltonian has the simple form [23, 24]

H = −t1∑i,j

(a†ibj +H.c.) (3)

where t1 ≈ 2.8meV is the nearest neighbour hopping parameter. The tightbinding eigen

functions has the form of bispinor, whose components corresponds to the amplitudes on

sublattice A and B respectively within the unit cell. This also means that in addition to its

usual spin 1/2, electrons carriers an additional pseudo-spin 1/2 associated with its sublattice

degree of freedom. The energy levels of the lattice can be determined by operating with the

Hamiltonian on its eigen function to give the corresponding band structure:

E = ±t1√

3 + f(k) (4)

where f(k) = 2 cos(qya√

3)+4cos(qx3a/2) cos(qya√

3/2) with lattice parameter, a ≈ 2.46A0.

In Eq.(4) the plus sign refers to the upper (π∗) band and the minus sign refers to the

lower (π) band. The unique feature of the band structure of graphene [25, 26] is that its

valence band (π) and conduction band (π∗) touch each other at 6 specific points, which are

also the vertices of the hexagonal Brillouin zone of this honeycomb lattice ( see Fig. 1). Out

of these six points, only two K and K ′ are non-equivalent and all the rest are related to

these two points by symmetry. These points are also known as the K points in the literature

and their positions in the reciprocal space are:

7

FIG. 1: (a) Honey comb lattice of graphene showing two inequivalent carbon atoms (with different

colour) in a unit cell (b) Band structure in honey comb lattice, with zoom in of the energy bands

close to one of the Dirac points.

K =

(2π

3a,

2π

3√

3a

),

K′ =

(2π

3a,− 2π

3√

3a

)Under ambient conditions, also known in the literature as zero biased doping, the Fermi

level coincide with these corners of the hexagonal Brillouin zone. Thus, to understand the

low energy charge transport properties of graphene one expands the full band structure close

to the K point by writing q = K + k, with k � K and then by keeping the first order

term and neglecting all higher order terms in k, the resulting dispersion relation is of the

form E = ±~vF |k| such that low energy quasiparticles of graphene are described by the

Dirac-like Hamiltonian [27–29]

H0 = vF (σ · p) (5)

where vF ≈ 106ms−1 is the fermi velocity, σ = (σx, σy) is the vector comprising of two

component Pauli matrices, and p = (px, py) is momentum vector in x − y plane. Thus,

charge carriers in graphene behave like massless relativistic fermions dubbed as massless

Dirac fermions with only the velocity of light is replaced by the Fermi velocity. This way,

the behaviour of transport electrons in graphene is very different from those in ordinary

semiconductors where they have a parabolic dispersion like a non-relativistic free particle.

8

In the following sections we shall describe how such charge carriers with ultra relativistic

dispersion laws transports in presence of highly inhomogenous magnetic field dubbed as

magnetic barriers in combination with local gate voltages.

II. ELECTRON TRANSPORT IN GRAPHENE : OPTICAL ANALOGY

A. Electron optics in the presence of scalar potential barriers

In a two dimensional electron gas (2DEG) it is well-established that transmission of de

Broglie waves satisfying the Schrodinger equation through a one-dimensional electrostatic

potential is similar to light propagation through a refractive medium. Such transport can be

understood in terms of phenomena like reflection, refraction and transmission, leading to an

analogy between electron transport and light propagation [1, 5, 6]. When a non-relativistic

electron in a 2DEG with quadratic dispersion, at Fermi energy E, is incident on a potential

barrier V , its momentum parallel to the interface outside and inside the barrier is conserved;

i.e., |~p1| sin θ1 = |~p2| sin θ2, where ~p1,2 are the momenta and θ1,2 are the angles they make

with surface normals respectively in the regions without and with potential barrier. This

leads to the following Snell’s law:

sin θ1

sin θ2

=

(1− V

E

) 12

(6)

Graphene charge carriers however do not have the quadratic dispersion, but instead

behave as massless Dirac-Weyl fermions leading to a different set of transport phenomena

[8–10, 22, 30, 31]. The optical analogues of such electron transport can again be constructed

by considering the charge carriers in MLG incident on an electrostatic potential barrier

V . Such charge carriers in monolayer graphene (MLG) obey the following Dirac-Weyl like

equation:

vF (σ · p) Ψ(x, y) = (E − Vi) Ψ(x, y) (7)

where Vi = V0 in the barrier region (|x| ≤ d) and vanishes outside, and Ψ(x, y) = [Ψ1,Ψ2]T

is the two component wavefunction with T denoting the transpose of the row vector. We

assume V = V (x) to be the one dimensional potential. Due to the y invariance of potential

barrier we consider the solutions of the form Ψ(x, y) = ψ(x)eikyy. On substituting this in

9

Eq.(7), two coupled equations in ψ1 and ψ2 are obtained as:

− i[d

dx± ky

]ψ2,1 = (E − Vi)ψ1,2

The above two off-diagonal equations can be decoupled in terms of ψ1,2 to give a

Schrodinger like equation of the form

[− d2

dx2+ k2

y

]ψ1,2 = (E − Vi)2ψ1,2 (8)

which admits exponential solutions,namely ψ1,2 ∝ exp(−ikxx).

By parameterizing the energy momentum relation k2xi

+k2y = (E−Vi)2 in polar coordinates,

we obtain

ky = E sinφ, kx1 = E cosφ, |x| > d,

ky = (E − V ) sin θ, kx2 = (E − V ) cos θ, |x| < d.

The momentum component along a straight interface should be conserved. Accordingly,

using the electron momentum conservation in the y direction at the left interface x = −d,

we obtain the Snell’s law in the following form:

(E − V ) sin θ = E sinφ

Clearly for E < V , the barrier acts like a medium with negative refractive index [11]. This

is shown in Fig.2

n =sin θ

sinφ= − E

|E − V |(9)

This corresponds to the electronic analogue of the well known phenomenon of negative re-

fraction, which occurs in left handed metamaterials and which was first proposed by Veselago

[12] and subsequently developed in more detail by Pendri and collaborators [13]. If one uses

so called split gate voltage by placing the region V < E and V > E side by side ( Fig. 2),

the Fermi level can be tuned below and above the Dirac point. Since the region below and

above the charge neutral Dirac points are respectively hole and electron states for such Dirac

fermions, this creates p and n type region in graphene simply by changing the height of of

the electrostatic potential barrier in a given region [11]. The resulting structure is called

graphene p−n junction. Consequently, a graphene p−n junction can be used as a Veselago

lense for electron focussing. Similarly circular graphene p-n junctions has also been studied

showing the formation of caustics [32].

10

FIG. 2: Schematic of the transmission through a potential step for an electron with (a) E > V (b)

E < V .

B. Klein Tunneling

While electrostatic potential barriers can indeed manipulate charge transport in graphene,

an electron could tunnel through any high barrier in contrast to the conventional tunnelling

of non-relativistic electrons [14, 33]. This behaviour, called Klein tunnelling in graphene,

leads to several observable transport effects related to transport some of which have been

demonstrated in graphene [34, 35] and also in carbon nanotubes [36]. Below we review

briefly the work on such tunnelling problem for graphene charge carriers in the presence of

an electrostatic potential barrier. This analysis will then be extended to similar transport

through inhomogenous magnetic fields and will be followed by comparing and contrasting

these transport to electrostatic and magnetic barriers, which is the central topic of this

review article.

The wavefunction solutions in the presence of a potential barrier can be obtained by

solving Eq.8. These solutions in any region of space can be written in terms of linear

combination of forward and backward propagating plane waves such that

ψ1(x) =

eikxx + re−ikxx, x < −d

aeiqxx + be−iqxx, |x| < d

teikxx, x > d

11

ψ2(x) =

s(eikxx+iφ − re−ikxx−iφ

), x < −d

s′(aeiqxx+iθ + be−iqxx−iθ

), |x| < d

steikxx+iφ, x > d

(10)

where kF = 2π/λ is the Fermi wavevector, kx = kF cosφ and ky = kF sinφ are the

wavevector components outside the barrier, qx =√

(E − V )2/~2v2F − k2

y ,θ = tan−1(ky/qx)

is the refraction angle, s = sgn(E) and s′ = sgn(E − V0).

By taking into account the continuity of wave-function components ψ1 and ψ2 at the

boundaries of the barrier the transmission and reflection coefficients are obtained as

t =2ss′e−ikxD cosφ cos θ

ss′[e−iqxD cos(φ+ θ) + eiqxD cos(φ− θ)]− 2i sin qxD(11)

r =2ieiφ sin qxD[sinφ− ss′ sin θ]

ss′[e−iqxD cos(φ+ θ) + eiqxD cos(φ− θ)]− 2i sin qxD(12)

Here D = 2d is the width of the barrier. Clearly qxD = nπ, n = 0,±1, ... corresponds

to the resonance condition at which the barrier transmission is unity. This corresponds to

usual resonant tunnelling through a potential barrier and occurs for non relativistic electron

as well. However remarkably, the above expression also shows that barrier always remains

transparent for normally incident carriers, ie. T = 1 when φ = 0. This feature of anomalous

tunnelling in graphene is peculiar to the Dirac like spectrum for graphene charge carriers

and is explained as follows: As explained briefly in section I A, the gapless, conical spectrum

of graphene is the result of intersection of two cosine like energy bands originating form

sublattices A and B. (represented in green and red for A and B sublattice respectively, see

fig.1). Due to this, the sublattice degree of freedom i.e. pseudospin should remain fixed on

each branch. This also means that an electron with energy E and possessing wavevector k

will originate from the same branch as a hole with energy −E and possessing wavevector

−k. Above two features allows the introduction of chirality for graphene charge carriers.

The term (σ · p) present in the Hamiltonian (5) gives the chirality, namely projection of

pseudospin on the direction of motion,

C =σ · p|p|

and has two possible eigen values, +1 (for electrons) and −1 (for holes). This operator is

same as the usual helicity operator for 3+1-dimensional relativistic electrons that obey Dirac

equation [37], but redefined for charge carriers in graphene which obeys 2 + 1 dimensional

12

Dirac-Weyl equation. In the absence of any potential, the chirality operator commutes with

the Hamiltonian and is therefore a conserved quantity. For normally incident carriers, the

chirality remains conserved even in the presence of an external electrostatic potential V (x)1.

This can be shown as follows: For normally incident carriers,

Chirality,C =σxpxpx

= σx

Hence

[C, H] = [σx,kxσx + V(x)1] = 0

Also the velocity operator

vx = −i[x, H] = σx

Thus the velocity operator is same as the chirality which is a conserved quantity for

this case [24, 38]. The velocity along the x direction is therefore a constant of motion

and thus cannot be reversed. This leads to perfect transmission through such a barrier

at normal incidence. This is what describes Klein tunnelling in MLG [14]. This is to

be noted here that a rectangular electrostatic barrier in graphene assumes that the gate

voltage-induced doping changes abruptly at the edges . In reality, however, the doping level

continuously changes and thus the edge of the potential barrier is actually smooth and not

sharp. This issue has been considered for scalar potential barrier in Ref.[39], where it was

found that a potential which is smooth on the scale of the Fermi wave length for small angles

of incidence, T (φ) = exp(−πkFD sin2 φ) where D is the barrier width. A comparison with

the transmission expression given in Eq.11 shows that the assumption of the rectangular

barrier captures the effect of Klein tunnelling correctly. However, at other angles close to

the normal incidence it overestimates the transmission. Transmission through trapezoidal

barrier was also analysed in Ref.[40] which combines the effect of a smooth barrier and a

rectangular barrier.

On expanding φ in eq.(12), close to the normal incidence, namely φ = 0, by substituting

φ = δφ0, it can be seen that the reflection amplitude r undergoes a π phase jump when the

incident angle φ goes from positive to negative value . At zero magnetic field, at non-normal

incidences, the two consecutive reflections on the two p-n interfaces occur with opposite

angles θ1 = θ and θ2 = −θ (Fig.20). When such a system is placed in a transverse magnetic

13

field B = Bz, the electronic trajectories bend in the presence of magnetic field. And above a

critical field value B > Bc, trajectory bending becomes sufficient to make the two consecutive

reflections occur with the same incident angle θ1 = θ2 which is exactly the case what happens

at normal incidence in the absence of magnetic field. This suddenly adds π to the phase

accumulated by an electron between two reflections and shifts the interference fringes by

half a period [41]. The observation of this half-period shift in Fabry Perot interference

fringes is therefore a direct evidence of perfect tunnelling at normal incidence [34]. We

analyse this in more detail in section IV D. An alternative way of demonstration of Klein

tunnelling in graphene p-n junctions has been addressed in [35]. This was done by probing

the transition from clean to disordered transport across a single steep p-n junction. Very

recently, an angle dependent carrier transmission probability in graphene p-n junctions has

also been investigated experimentally [42] and theoretically [43] where it is shown that chiral

tunnelling can be directly observed from the junction resistance of a tilted interface probed

with separate split gates.

III. ELECTRON TRANSPORT IN THE PRESENCE OF INHOMOGENOUS

MAGNETIC FIELD PROFILE

While the above mentioned absence of backscattering by a scalar potential due to Klein

tunnelling is a very interesting phenomenon, this implies that confining transport electrons in

graphene by a potential barrier is not possible in a conventional way. This leads to problem

in device making. For example, even though it is easy to make a p−n junction in graphene,

reverse biasing such junction will be very difficult for Klein tunnelling. For the designing of

graphene based electronics it is crucial to attain confinement of electrons within a mesoscopic

or nanoscopic size of the sample. For this reason, several alternatives have been suggested.

One way is to exploit the fact that suitable transverse states in a graphene strip may allow

one to circumvent Klein tunnelling [44]. Other schemes that have been proposed include

gated nanoribbons [45], gated [46] or doped [47] bilayer etc. Yet another possibility was

demonstrated theoretically in [16],[48] by making use of external inhomogeneous magnetic

fields applied perpendicular to the graphene plane. This brings us to the the main focus

of this review article- to investigate the electron transport in the presence of inhomogenous

magnetic field viz. magnetic barrier(s) in graphene. Before proceeding further we first

14

FIG. 3: Magnetic field under the stripe corresponding to the (a) perpendicular (b) parallel config-

urations.The magnetic field is given at the following distances from the magnetic stripe: z0 = 0.15

(solid curve), z0 = 0.3 (dashed curve), and z0 = 0.5 (dotted curve).

discuss the experimental strategies for creating such inhomogenous magnetic fields and point

out that such magnetic barriers are already in very much use in other materials.

A. Producing inhomogenous magnetic field profile: Experimental strategies

A microscopically inhomogenous magnetic field can be created for charge carriers in

graphene by placing a graphene sheet in close proximity to long magnetic stripes that

produce highly localized magnetic fields. Such field profiles can be generated using de-

magnetizing fields produced at the edges of narrow stripes made with hard ferromagnetic

(FM) materials of either perpendicular or in-plane anisotropy. The magnetic field of such

structures is given as

B = B(x, z0)z = B0[K(x+ d, z0)−K(x− d, z0)]z

where K(x, z0) = − 4xdx2+z2

0for perpendicular magnetization, and K(x, z0) = − 2z0d

x2+z20

for mag-

netization parallel to the graphene sheet. B0 is a constant dependent on the aspect ratio of

the stripe.

For a given value of z0 we plot in Fig.3(a), (b) the profile of such a magnetic field. The

15

major component of the demagnetizing field reaches the graphene monolayer and produces

the desired field profile. Though there is always some component of the demagnetizing field

that will give rise to undesired fringe fields in other directions, these can be substantially

lowered by suitable magnetic design of the stripes. Such nanostructures are routinely used

in magnetic recording media [49]. Materials such as CoCrPt produce fields of 1 Tesla close

to the surface with bit lengths ranging from 50-100 nm. For example, in [50], isolated tracks

of single-domain magnetic islands have been fabricated using focused ion-beam lithography

for both perpendicular and parallel anisotropy using CoCrPt. Here, patterns with successive

magnetizations pointed along opposite directions were achieved. In another recent work, off-

axis electron holography has been used to probe the magnetization structure in high density

recording medium by using perpendicular magnetic anisotropic (PMA) recording medium

[51]. The direct imaging of magnetization done shows that the foils of PMA material consist

of successively reversed highly stable domain structures of few ten’s of nanometer size. In

practice, one can also change the strength of the magnetic field by suitable adjusting the

width of such PMA material. Precise design of read-write structures for recording individual

bits at these dimensions has also been achieved [52]. Using the above discussed techniques,

typical magnetic barriers can be patterned down to 50-100 nm widths.

Apart from the above mentioned method of using ferromagnetic materials [53] there are

several other ways in which inhomogenous magnetic fields have been experimentally realised

in conventional semiconductors. One such way is through the integration of superconducting

elements, which may be used to screen the externally applied magnetic field in accordance

with Meissner effect. When an external magnetic field is applied the flux lines will be ex-

pelled from the superconductor due to the Meissner effect. If the latter is close enough

this will result in an inhomogeneous magnetic field in the 2DEG [54][55] . This is shown

schematically in [Fig.3(d)]. As an example, in ref. [54], it was shown that with type II super-

conducting films deposited on top of the two-dimensional electron gas in a GaAs/AlGaAs

heterostructure, the distribution at the 2DEG takes the form of flux tubes which are much

narrower than an electron inelastic scattering length. The superconducting materials used

in these experiments were typically 200nm of lead (Pb), 400 nm of a lead/indium [Pb(1 at.

% In)] alloy, and 200 nm of niobium nitride (NbN), as being materials whose characteristic

superconducting length scales span a broad range. Growing the heterojunction on top of a

pre-etched (nonplanar) substrate can also give rise to inhomogeneous magnetic fields [56]-

16

FIG. 4: (color online)(a) Cobalt finger gates at the surface of a 2DEG. (b) Magnetic modulation

profiles at different tilt angles of the magnetization.(c) MFM image of the cobalt grating magnetized

along the short axis of the stripes, in the plane of the 2DEG. The magnetic poles appear as the

black lines. (d) MFM image of the grating magnetized along the long axis. The magnetic poles

are absent. Parameters: a = 400nm, d = 200nm, h = 160nm and z0 = 90nm. [This figure is taken

from [17], printed with permission.]

[58]. When a uniform external field is applied on such a system, the angle between the field

direction and the normal to the 2DEG depends on the tilt of the facet and therefore will

have a normal component of magnetic field which varies spatially across the sample. Since

it is only the normal component of magnetic field that influences the transport, by varying

the facet length and angle, as well as the angle between the applied field and the normal to

the substrate, a wide variety of field profiles can be generated.

In other studies[59]-[61] a similar yet simpler approach was used based on the idea that

if a thin magnetic film is placed on top of a heterostructure its in-plane magnetization can

be saturated in an external in-plane magnetic field [Fig.3(c)]. In this case the out-of-plane

component of the fringe field under the edge of the film creates a magnetic barrier for electron

transport with a width of the order of 100 nm determined by the separation of the 2DEG

from the magnetic film. Magnetic field strengths of more than 0.5 T have been realized in

this way [61]. A detailed discussion on experimental methods used for applying magnetic

17

FIG. 5: Transmission in the presence of finite width magnetic barrier in monolayer graphene,

D = 2d = 1.5lB (a)B = 0.5T (b) B = 3T .

modulations to 2DEGs can be found in the review article [17].

B. Magnetic confinement of massless dirac fermions in graphene

In the previous section we reviewed the experimental progress of realizing inhomogenous

magnetic field or magnetic barrier over a wide range of length scales in a number of sys-

tems. Mesoscopic transport in presence of inhomogenous magnetic fields has been studied

theoretically [62], [63] - [64] for non relativistic electrons . Related experimental studies also

took place in conventional semiconductor heterostructures, e.g., transport in the presence of

magnetic barriers [65] and superlattices [66], magnetic edge states close to a magnetic step

[67], and magnetically confined quantum dots or antidots [68]. In the following section B we

review the transport of massless Dirac fermions in graphene in the presence of of magnetic

barriers with some suitable examples.

The magnetic barrier profile chosen here is in the form of step like function [16, 48] as

B = BΘ(d2 − x2)z (13)

, and the vector potential chosen in Landau gauge Ay(x) = [−Bd,Bx,Bd] in region x < −d,

|x| < d and x > d respectively.

Typically, the equation of motion of charge carriers at the two inequivalent Dirac points

K and K ′ are decoupled from each other [69] in the absence of intervalley scattering which

gives a wave vector change of the order of 2kF . We assume for the cases to be considered

18

this condition is satisfied. Under this situation motion near each Dirac point can be treated

separately. The stationary solutions near each such point is given by a two component spinor

and satisfies the equation

vF (πx ∓ iπy)Ψ2,1 = EΨ1,2 (14)

Here, vF is the fermi velocity (vF ≈ c/300) and π = p + |e|cA. Using lB as the unit of length

scale such that x = xlB

and ~vFlB

as the unit energy such that ε = ElB~vF

and Ψ(x, y) = ψ(x)eikyy

in the Landau gauge, Eq.14 can be rewritten as

− i[∂

∂x±(kylB +

Ay(x)

BlB

)]ψ2,1 = εψ1,2 (15)

which can be decoupled to obtain the following Schrodinger like equation:

[− ∂2

∂x2+ (kylB + x)2 + 1

]ψ1,2 = ε2ψ1,2 |x| < d[

− ∂2

∂x2+ (kylB ∓ d/lB)2

]ψ1,2 = ε2ψ1,2 (16)

where (−/+) sign in the above expression corresponds to region (x < −d)/(x > d) respec-

tively.

Upon solving, we obtain the solutions outside the barrier as propagating solutions while

inside the barrier the solutions are localised and can be written in terms of parabolic cylin-

drical functions [70].

ψ1(x) =

eikxx + re−ikxx, x < −d

aDε2/2−1(√

2(x+ kylB))

+bDε2/2−1(−√

2(x+ kylB)), |x| < d

teik′xx, x > d

(17)

ψ2(x) =

ei(kxx+φ) + re−i(kxx+φ), x < −d

a i√

2εDε2/2(

√2(x+ kylB))

+b−i√

2εDε2/2(−

√2(x+ kylB)), |x| < d

tei(k′xx+φ′) x > d

(18)

where kx = kF cosφ, k′x = kF cosφ′, D stands for parabolic cylindrical function, and φ

and φ′ corresponds to the incidence and emergence angle in regioin I and III respectively.

19

The emergence angle can be obtained by conservation condition of ky as:

sinφ′ =2d

εlB+ sinφ (19)

An alternative way of writing the above equation is

ε sinφ′ = 2πΦ

Φ0

+ ε sinφ

where Φ0 = hc/e and Φ = 2BdlB, is the net flux enclosed by an area 2dlB. The second

form particularly shows how the flux enclosed by the area containing the barrier changes

the refractive index of this region in a non specular way.

The transmission through such magnetic barrier can be obtained by matching the wave-

function at the boundaries of the barrier, T = kxk′x|t|2, where the prefactor ensures the current

conservation in the system. The transmission as function of incidence angle φ and incident

energy E is shown in fig.5. As one can see, Eq. 17 and 18 depicts the formation of lo-

calised states inside the barrier region. However these states are bound only in the direction

perpendicular to the barrier but can propagate in the direction along the inhomogeneity of

the magnetic field. The transmission, for this reason, depends very strongly on the incident

wavevector and facilitates the possibility to construct wave-vector filter [73]. Fig. 5 depicts

this situation clearly. The requirement of the propagating solutions in the incident region

and region of exit determines the range −π/2 < φ < sin−1(

1− dεlB

)beyond which no trans-

mission is possible, thereby leading to the possibility of confinement. This is how magnetic

barrier turn reflective for a set of wave vector for the massless Dirac fermions. In the follow-

ing section we shall illustrate this property more clearly by building clear optical analogy

of electron transmission through the extremely inhomogeneous magnetic barrier where the

magnetic field profile can be approximated as a delta function.

IV. MASSLESS DIRAC FERMIONS IN INHOMOGENOUS MAGNETIC FIELDS:

AN ANALOGY WITH LIGHT PROPAGATION

A. Magnetic vector potential (MVP) barrier

The issue addressed here is whether the transport through such magnetic barrier can be

understood in terms of propagation of light through medium with changing dilelectric con-

stant using well known ideas in geometrical optics[19]. Even though such idea was developed

20

for the ballistic electron transport through electrostatic potential barrier, generalizing this

concept in presence of a magnetic field was non trivial since in the presence of magnetic field

electron executes cylotron motion and the wave vector k is not a good quantum number for

such motion. Thus to yield a Snell’s law like one in Eq. (9) in presence of a magnetic barrier,

it is essential that the electron should not complete its half cyclotron radius beyond which it

get completely localised inside the barrier. This situation can be realized in the presence of

a highly inhomogeneous magnetic field. Particularly, if the range of inhomogeneity is much

smaller than the cyclotron radius, one is left with plane wave-like scattering states. Thus,

such field profile scatter the electrons in the same way as an electrostatic potential does.

For this to be valid, two conditions must be satisfied. First, the magnetic length lB =√~ceB

should be similar in order to the width of such magnetic barriers. Second, the de

Broglie wavelength λF should be much larger than the barrier width so that the electron

will not see the variation in the vector potential inside the barrier. An extreme case of

such an inhomogeneous magnetic field is singular magnetic barrier - the one introduced in

[19–21, 64, 71–73], which gives rise to step-function like magnetic vector potential (MVP)

barriers. Here, we shall consider the scattering of massless Dirac fermions by such singular

magnetic barriers as in Eq.20. Such field profiles having highly localized field variations are

well known in literature and have been realized experimentally as we have already pointed

out in the section III A. Patterned stripes down to length scales as low as 10 nm have

been realized using nanolithography [74]. It is possible to achieve field profiles at even

smaller dimensions using domain walls with widths in the range of 10-50nm and magnetic

nanostructures down to 5 nm [75, 76] and 0.15 nm [77] having highly localized field variations

which can be approximated as such delta function barriers.

1. Electron optics with MVP barrier

The most elementary of such magnetic field profile that satisfy the flux line closure

condition and its vector potential in the Landau gauge can be given as :

B = BlB[δ(x+ d)− δ(x− d)]z (20)

Ay(x) = BlBΘ(d2 − x2)y

21

FIG. 6: Monolayer graphene with ferromagnetic stripes having magnetizations perpendicular (a)

and parallel (b) to plane. The magnetic field B (red) produces a magnetic vector potential A

(blue). Single MVP barriers are formed in (a) and (b).

In the absence of intervalley scattering, the charge carriers in graphene in the presence of

above potential form can be described by Eq.14 which can be rewritten in the form of the

following coupled equation

− i[∂

∂x± (kylB + ∆)

]ψ2,1 = εψ1,2 (21)

Here ∆ = 1 for |x| < d and ∆ = 0 for |x| > d. The coupled equations Eq.21 can be

decoupled easily and results in a Schrodinger like equation of the form[− ∂2

∂x2+ (kylB + ∆)2

]ψ1,2 = ε2ψ1,2 (22)

Thus, in the barrier region −d < x < d electrons see a momentum-dependent barrier of

height [ky − sgn(e) 1lB

]2. Hence such a barrier will be termed as magnetic vector potential

barrier (MVP).

Since magnetic field doesn’t do any work, the energy conservation gives k2x + k2

y = k2F for

|x| > d and q2x + (ky + 1

lB)2 = k2

F for |x| < d. By parameterizing the above two equations in

polar co-ordinates as shown in Fig. 7, we obtain the relation

sin |θ| = sin |φ|+ sgn(φ)1

kF lB,−π/2 < φ < π/2 (23)

The situation is depicted in Fig.7. For a wave incident with positive φ wave vector bends

away from the normal whereas for a wave incident with negative incidence angle the cor-

responding wave vector bends towards the surface normal inside the barrier region. Here,

22

FIG. 7: Asymmetric refraction through a single MVP barrier.

θ is the angle of refraction (denoted as θ1,2 for positive and negative angles of incidence).

Clearly, the Snell’s law for electron waves in such magnetic barriers is not specular as it is

for light waves on smooth surfaces or for the incidence of the electrons on an electrostatic

potential barrier [11], [14].

According to Eq.23, when |sin |θ|| > 1, θ becomes imaginary and the wave in the second

medium becomes evanescent. In the language of optics this corresponds to total internal

reflection (TIR). According to Fig. 8, this will happen when sin |θ| > 1 for 0 < φ ≤ π2

and when sin |θ| < −1 for −π2≤ φ < 0. In the latter case, this requires the wave vector

to be negatively refracted [39] at sufficiently high magnetic field before TIR occurs. It also

follows that for a given strength B the magnitude of critical angle of incidence |φ| = φc for

TIR is higher for −π2≤ φ < 0 as compared to the one for 0 < φ < π

2. Because of TIR

the transmission on both sides of Fig 8 drops to 0 beyond a certain value of φ and for fixed

B field this value is higher for negative angles of incidence . Thus we can understand the

reflective nature of such barrier using the corresponding Snell’s law given in 23. To illustrate

this we shall now calculate the transmission through such barrier.

2. MVP barrier: Transmission probability

On solving Eq.22, the corresponding wavefunctions in any region of space can be written

in terms of a linear superposition of forward and backward moving plane waves. Then

23

FIG. 8: Transmission T as a function of φ and E for single MVP barrier (a) B = 0.5T , d = 5lB

(b) B = 3T , d = 20lB.

continuity of the wavefunction at the boundaries of the MVP barrier can be used to calculate

the transmission coefficient as

t =2ss′e−ikxD cosφ cos θ

ss′[e−iqxD cos(φ+ θ) + eiqxD cos(φ− θ)]− 2i sin qxD(24)

Here D = 2d. And s and s′ are given by sgn(ε) and are both +1 for electrons when

only magnetic fields are present. It may be pointed out that the form of the transmission

expression is same as the corresponding one for electrostatic potential barrier (11). But

because the wave vector inside the barrier is different as compared to those in presence

of scalar barrier, the transmission profile is fundamentally different. Fig.8 shows how the

Transmittance, T = t∗t changes with the angle of incidence. As suggested by Eq.23, the

transmission is clearly asymmetric unlike the case of electrostatic potential [14].

For high electrostatic barriers such that V >> EF , the wavevector is given by qx =√(EF−V )2

~2v2F− k2

y, which is real. The corresponding transmission for electrostatic potentials is

T =cos2 φ

1− cos(qxD) sin2 φ

and is 1 at φ = 0, exhibiting Klein tunnelling for massless Dirac fermions [14]. In contrast

at high values of the magnetic barrier since 1kF lB

∝√B, the wavevector given by q2

x =

k2F − (ky + 1

lB)2 = −κ2 < 0. This leads to TIR and not Klein tunnelling. As seen in Fig. 8,

the magnitude of critical angle beyond which TIR occurs is lower for a higher magnetic field.

Thus, a stronger MVP barrier leads to higher reflections as opposed to complete transmission

24

at normal incidence by high electrostatic potential barrier. Complete transmission occurs

only for qxD = nπ as given by Eq.24. This corresponds to resonant tunnelling for Dirac

electrons and happens in the same way as for nonrelativistic electrons, appearing as a number

of peaks in the plots in Fig.8 . The number of tunnelling peaks increases with barrier width

for both MVP and electrostatic barriers.

In conclusion, we see that the reflectance can be controlled by suitably modifying the

strength and locations of the magnetic barriers and thereby changing the refractive index of

the intervening medium in a novel manner. This principle could be the basis of structures

like magnetic waveguide, where the reflection must be high at desired propagation angles,

here, this could be manipulated by changing the magnetic field. Also for strutures like

the resonant cavity, high reflection is needed near normal incidence. Geometries such as

three-mirror or four-mirror cavities [19] could be used for high reflection at other angles .

B. Combined MVP barrier and electrostatic potential (EMVP) barrier

The charge carriers in graphene have a linear band structure albeit only close to the

Dirac point, coincident with the Fermi level EF in undoped graphene. Small electrostatic

potentials greatly affect electron states by shifting the Dirac point with respect to EF and

causing the graphene sheet to behave as either an electron-deficit (p-type) or a hole-deficit

(n-type) material. This situation was also experimentally verified by observing electron hole

puddles [15, 78] near the Dirac point in monolayer graphene. Thus, the effect of electrostatic

potentials on any proposed graphene structure must be included.

Quantum hall effect in gate controlled p-n junction in graphene in presence of uniform

magnetic field was already studied experimentally [79] which reveals new quantum hall

plateaus. The top gate geometry was utilized in controlling the edge channels in the quan-

tum Hall regime and with control over local and global carrier density. The effect of local

modulation of charge density and carrier type in graphene field effect transistors using a

double top gate geometry has also been studied [80]. Also, in a recent experiment by S.

Pisana et al.[81], the enhanced magneto-resistance of a monolayer graphene sheet has been

measured by connecting it to two voltage and two current terminals and simultaneously

exposing it to various magnetic field strength at room temperature. The differential voltage

as a function of the magnetic field has been plotted thereby analysing the joint effect of

25

magnetic field and the applied voltage on the magneto-transport particularly close to the

Dirac point. From the analysis of the magneto resistance data it was inferred that the band

structure of transport gets strongly modified in presence of voltage and magnetic field. The

above experiment result clearly shows that simultaneous application of voltage and magnetic

field strongly influences band structure, as we’ll also see in the following section. For most

of the above experiments the magnetic field applied is homogenous over the typical size of

the sample. A complimentary case where as for our case we consider magnetic barrier which

is a highly inhomogenous magnetic field over the typical size of the sample.

FIG. 9: EMVP barrier structures for graphene. In (a)-(c), the magnetic field is applied using

patterned ferromagnetic (FM) lines with perpendicular anisotropy and two potentials +Vg and

−Vg are applied by separate conductor lines. As given in the text, effective potential induced in

the graphene is V . In (b), the magnetic strength of the middle line is doubled by either using a

different FM material or by larger dimensions. In (d), the magnetic field is produced from the two

edges of a FM stripe with in-plane anisotropy and the same stripe is also used to apply one of the

potentials.

We first discuss briefly some of the practical issues when a electrostatic gate potential is

applied to the graphene sheet. A gate voltage ±Vg can be applied using a metal electrode

and separating the electrode from the graphene layer with an insulating oxide layer. This

26

oxide layer which serves as a dielectric medium between graphene sheet and metal electrode

can be made as thin as several nanometers. When such a voltage Vg is applied locally, it

induces electron (hole) doping ±σn proportionally and changes the carrier concentration

within the channel. This shifts the undoped Fermi level EF from the Dirac point by an

amount V = sgn(σn)√σn~vF , where sgn(σn) is the sign of the induced charge. Due to

this a local potential barrier V is created which is equal to the difference between the local

Fermi level and the Fermi level in the undoped region. Since the typical breakdown strength

of dielectrics such as alumina and fused silica are around 10 − 20MV/m, a graphene layer

could be subjected to 1V applied across a 100nm thick dielectric. Following Refs.[8, 9], gate

voltages Vg of upto ±100V have already been applied to graphene flakes. The electrostatic

gate potentials would be generally applied by separate conductors placed suitably in different

planes than the FM stripes, but in some cases, the FM stripes could also be used to apply

voltages.

1. Electron optics with EMVP barrier

Below we discuss the transport in the presence of a single MVP barrier and a commen-

surate electrostatic step potential (EMVP) barrier.

V (x) = V,Ay(x) = BlB, |x| < d

V (x) = 0, Ay(x) = 0, |x| > d (25)

We assume V > 0 and set the incident energy at E = EF , where EF is the Fermi level located

at the charge-neutral Dirac point in monolayer graphene and the voltage V is measured with

respect to EF . The entire treatment that follows is applicable in the neighbourhood of the

charge-neutral Dirac points as long as the dispersion remains linear. In addition, we also

assume that scattering by an EMVP barrier is not strong enough to break the degeneracy

of the K and K ′ points. In the region −d < x < d at either of the K points, the motion will

be described by

vF

V/vF πx − iπyπx + iπy V/vF

ψ1

ψ2

= E

ψ1

ψ2

(26)

27

FIG. 10: (a) Function SF as a function of voltage (b) Ray diagram of electron propagation through

a single EMVP barrier. With increasing V , the refracted angle θ2 increases continuously in the

clockwise direction. Dashed rays are negatively refracted waves.

Here π = p+ ecA. In the Landau gauge, the stationary solutions can again be written as

ψ1,2(x, y) = ψ1,2(x)eikyy

Substituting this in Eq.26, we get the following coupled equations 0 −i∂x − i(ky + 1lB

)

−i∂x + i(ky + 1lB

) 0

ψ1

ψ2

=E − V~vF

ψ1

ψ2

(27)

The above equation can be decoupled to yield[−∂2

x + (ky +1

lB)2

]ψ1,2 =

(E − V~vF

)2

ψ1,2 (28)

The corresponding stationary solutions ψ1,2(x) are

ψ1 =

eikxx + re−ikxx if x < −d

aeiqxx + be−iqxx if |x| < d

teikxx if x > d

(29)

28

FIG. 11: Phase diagram of refraction angles θ as a function of V on x-axis and φ on y-axis, and

for an MVP barrier of width D = 100nm at B = 0.1 and 3T . The gap corresponds to the region

where TIR occurs.

ψ2 =

s[ei(kxx+φ) − re−i(kxx+φ)] if x < −d

s′[aei(qxx+θ) − be−i(qxx+θ)] if |x| < d

stei(kxx+φ) if x > d

(30)

These are similar in form to those for a pure magnetic barrier or an electrostatic step

potential 10. But (kx, ky) and(qx, ky + 1

lB

), namely the x and y components of the wavevec-

tor inside and outside the barrier regime are different. Here, s, s′ are sgn(E − V ) in the

respective regions. Upon setting incident energy EF as ~vFkF , substitution of Eqns.(29)

and (30) in Eq.(28) leads to

k2x + k2

y = k2F , with kF =

EF~vF

, |x| > d

q2x + (ky +

1

lB)2 = (kF −

V

~vF)2 = k′2F , |x| ≤ d (31)

The incidence angle φ and the refraction angle θ are given by tan−1(kykx

) and tan−1(kylB+1

qx lB)

respectively. Eq.(31) can then be rewritten to obtain the Snell’s law analogue for electron

29

waves of such massless Dirac fermions incident on the EMVP barrier as

sin |θ| = SF (sin |φ|+ sgn(φ)1

kF lB) (32)

= SF sin |θ||V=0

SF =kFk′F

= s′| EFEF − V

|; s′ = sgn(EF − V )

Comparison between Eq.(23) and Eq.(32) shows that the potential barrier effectively

scales the refraction angle by the scale factor SF defined above. SF is a non-monotonic and

discontinuous function of V [Fig.10(a)] and we shall study its impact on the refraction of the

incident electron wave. There are basically two regimes to be analysed depending upon the

strength of electrostatic potential applied (i) V < EF , (ii) V > EF . For positive incidence

angles φ = (0, π/2), in the absence of any electrostatic potential the refraction angle |θ| is

larger than the incidence angle |φ|, thus the electrons are seen as passing from denser to rarer

medium. As V increases from 0 to EF , the function SF and hence θ increases continuously

(this is shown in fig.10(a) and (b) respectively) and eventually the electron wave suffers

TIR as soon as the r.h.s of Eq. 32 becomes greater than 1. Thus, by increasing V it is

possible to totally reflect an electron wave for any given φ and B. Since the reflectivity of a

magnetic barrier increases with higher B, this implies that the addition of V can effectively

convert a weaker magnetic barrier into a stronger one. This effective control of the strength

of the magnetic barrier by an electrostatic potential ( gate voltage) is a consequence of the

ultrarelativistic nature of the charge carriers in graphene. In a subsequent section we shall

provide a more formal reasoning for this feature.

When V surpasses EF , the sign of sin |θ| will be opposite to the sign of sin |φ|. At smaller

V close to EF there is still TIR, whereas for very high V much above EF the refraction

becomes negative and the wave again retraces its path back in the barrier regime. This

situation is depicted in Fig. 10(b), where the wavevector for the refracted ray is shown

changing with increasing V. A similar analysis can be done for negative incidence angles.

We summarize this discussion by plotting sin |θ| as a function of the incidence angle φ for

different V and B values in Fig. 11. As can be seen, each of Fig.11(a) and (b) are separated

into an upper and a lower part by the line φs = sin−1(− 1kF lB

). In the lower part where |φ| is

larger than |φs|, with increasing V the absolute value of the refracted angle, |θ| exhibits four

phase regions: rarer → denser, denser → rarer, TIR gap, negative refraction. In the upper

part, the four phase regions exhibit a different order with increasing V : negative refraction,

30

FIG. 12: Here kx and ky represent the transverse and longitudinal components of the wave vector.

The circles represent solutions of the dispersion relations in the domains 1 and 2 and correspond

to propagating waves.[82]

TIR gap, denser → rarer, rarer → denser. All these different regions meet at the limiting

point (EF , φs), where the behaviour is singular and is discussed later on in this article in

more detail.

The refraction laws for the electron waves incident on an MVP/EMVP barrier can be

alternatively understood with the following geometric consideration[82]. We consider two

domains with scalar and vector potential (V1, A1) and (V2, A2) respectively. The electron

wave being incident from domain 1 at an angle φ and refracted in domain 2 at angle θ. Then

the parallel and perpendicular wavevector components in domain 1 and 2 are related by the

following dimensionless dispersion relations:

k2x1,2

+ (ky1,2 +A1,2)2 = (ε− u1,2)2 (33)

Each of the eq. 33 represents a circle in the (kx, ky) plane, so that the wavevector for

the propagating solutions will lie on the circle itself. Now, the refraction laws can be easily

31

derived from fig.12 as follows. Due to translational invariance along the y-direction, the

perpendicular wavevector component takes the same value in both domains ky1 = ky2 =

ky0 . For a given ky0 , propagating solution kx in domain 1 and 2 will take the value which

lie at the intersection of line ky0 = constant with the corresponding circles. There are

two such intersections for each of the circle. (This is shown in figure at O1,2 and O′1,2).

The physically meaningful intersection will be determined by the sign of (ε − u1,2). For

definiteness we assume the incident current density to be positive i.e. kx1 > 0. Then

using the parametrization kx1 = (ε − u1) cosφ, cosφ will be positive or negative when

(ε − u1) > or < 0 respectively. This means the incident wave will be directed along the

vector joining the centre of circle with O1 when (ε − u1) > 0 or opposite to the vector

joining the centre of circle with O′1 when (ε− u1) < 0. Similarly, using the parametrization

kx2 = (ε−u2) cos θ, the refracted current (which should possess the same sign as the incident

current density) will be directed along the radius vector from the centre of the circle to the

intersection point O2 when (ε − u2) > 0, or opposite to the radius vector from the centre

of the circle to O′2 when (ε − u2) < 0. Simple geometry from the fig.12 gives the following

relation that defines the connection between the incident φ and refracted θ angles

ky0 = −A1 − (ε− u1) sinφ = −A2 + (ε− u2) sin θ

This relation gives the same condition as in eq. 32 for EMVP barrier.

2. EMVP barrier: Transmission probability

Transmission through a magnetic barrier gets strongly affected due to the presence of V .

Again from the continuity at x = ±d in Eqs.(29) and (30), the transmission coefficient can

be obtained as

t =2ss′e−ikxD cosφ cos θ

ss′[e−iqxD cos(φ+ θ) + eiqxD cos(φ− θ)]− 2i sin qxD, (34)

where D = 2d and qx = k′F cos θ. Here s and s′ corresponds to sgn(E) and sgn(E-V)

respectively. The expression for t is same as the one for electrostatic potential barrier 11 or

MVP barrier. However the difference in the result is due to the change in the expression for

qx, namely the x-component of the wave vector inside the barrier regime. In fig.13 is plotted

transmittance T = t∗t as a function of φ and V in the presence of an EMVP barrier. When

32

FIG. 13: Transmission in the presence of EMVP barrier in monolayer graphene, B = 1T , EF =

50meV , D = 100nm.

the potential is lesser than but close to Fermi energy, transmission takes place over a very

small window located asymmetrically along the φ axis. As the point V = EF is crossed, the

function SF changes sign thereby changing the sign of refraction angle. Thus, the potential

barrier induces more asymmetry in the transmission as against pure MVP barrier, this is

clearly depicted in Fig. 13.

The point V = EF represents a singularity in the spectrum and demands a separate

discussion. In the absence of magnetic barriers, such a point represents the zero modes for

Dirac operators and leads to the emergence of new Dirac points. This has been discussed in

a number of recent works considering Andreev Reflection in a graphene based NIS junction

[83] and for electronic states of graphene in a periodic potential [85–87]. The presence of a

magnetic barrier breaks the time reversal symmetry explicitly and these zero modes become

the zero modes of the modified Dirac operator and the corresponding solutions are different.

The equations satisfied by ψ1,2 are

[∂x − (ky + 1/lB)]ψ1 = 0

[∂x + (ky + 1/lB)]ψ2 = 0

for which the solutions are obtained as

ψ1 = c1e(ky+1/lB)x

ψ2 = c2e−(ky+1/lB)x

33

FIG. 14: Averaged transmission Vs voltage through a single EMVP barrier in monolayer graphene

for different magnetic fields.

By using the continuity condition at x = ±d, the transmission expression is obtained to be

t =2 cosφe−ikxD

eiφe(ky+1/lB)D + e−iφe−(ky+1/lB)D

The significance of such zero modes is that on the two sides of the singular point, the relative

sign between the φ and θ becomes opposite.

3. Conductance: Effect of various EMVP Barriers on Transport

To see how the above angle-dependent transmission properties effect electron transport

through such barriers, we plott the average transmission through the barrier as a function

of the potential V at various strengths of the magnetic barrier. The average transmission

at a given barrier strength V and B is defined as [73]

〈T (B, V )〉 = vF

∫ π2

−π2

dφT (φ,B, V ) cosφ (35)

This formula, when generalized to a range of energy levels, leads to the Landauer con-

ductance G ∝ 2πne2〈T (B, V )〉/h. This has been plotted for three different strengths of

magnetic barrier B over a range of potential barrier strength V for a single EMVP barrier

34

in Fig. 14. The transmission shows a minimum as expected when the point qV = EF is

approached. The transmission grows on both side of this singular point and finally at higher

|V | oscillates around an average value.

4. Generalisation to MVP/EMVP barriers of any arbitrary shape

A general algorithm for the calculation of electron transmission in graphene through

inhomogeneous electric and magnetic fields of any abitrary profile has been carried out in

[88] using transfer matrix method. The fields are invariant in one direction; and the method

involves the division of the one-dimensional domain into slices and taking an appropriate

approximation of potential form in each slice. The equation for each slice is solved and the

continuity conditions are used at the interfaces of two such slices. The exact solution of the

equation in each slice depends on the potential form chosen. It is shown that in the presence

of piecewise constant scalar potential and piecewise linear vector potential the resulting

equation admits parabolic cylindrical functions as the solution basis in moderate magnetic

fields while the basis functions tend to complex exponentials in the presence of extremely

small magnetic fields. Also it is shown that the localised charge distribution (electron and

hole puddles) which arises due to the presence of disorder in graphene [78] can be modelled

as the charge as arising from a scalar potential distribution proportional to it. With this the

transmission calculation corresponding to a one-dimensional potential extracted from the

experimental data in ref. [78] is presented. The transport analysis with magnetic barriers

in which the edges are smoothed out have also been carried out [89]- where a hyperbolic

profile has been chosen. Here the corresponding Dirac equation can be analysed within the

formalism of supersymmetric quantum mechanics, and leads to an exactly solvable model.

Also, exact solutions have been given for a Dirac electron in the presence of an exponentially

decaying magnetic field [90].

C. Quantum Goos-Hanchen Shift in single MVP and EMVP barriers

Analogues of optical phenomena such as refraction [11], collimation [91], Fabry Perot

interference [92], Bragg reflection [19] etc. in the ballistic transport regime for Dirac fermions

in mono and bilayer graphene based structures in presence of scalar and vector potentials

35

have been proposed. We discuss them in various sections of this review. In the following

section we discuss another important optical phenomenon, the Goos-Hanchen (GH) effect

[93, 94]. The GH effect is a phenomenon of classical optics that describes the lateral shift

between the centre of a reflected beam and that of incident beam when a total reflection

occurs at the interface between two media. The shift occurs as the totally reflected ray

undergoes a phase shift with respect to the incident beam, and this is is detectable since the

extent of a real beam is always finite. The shift reverses sign if the second medium behaves

like a metamaterial with negative refraction [95, 96]. Such a lateral shift for totally as well as

partially reflected electron waves can also occur for non-relativistic electrons passing through

a semiconductor barrier [97], magneto-electric semiconductor nanostructure [98]. Recently,

it has been shown that ballistic electrons passing through a p-n interface in graphene [99]

also suffer a GH shift, which changes sign at certain angles of incidence. The analysis was

extended for the case of MVP and EMVP barriers subsequently [21]. To understand GH

shift in graphene, we consider the following wavepacket (beam) of electrons impinging on a

p-n interface or a MVP/EMVP barrier at energy EF :

Ψin(x, y) =

∫ ∞−∞

dkyf(ky − k)eikyy+ikx(ky)x

1

eiφ(ky)

(36)

The envelope function ensures the wavepacket is of finite size along the y-direction and is

sharply peaked at ky = k. Thus, k ∈ (0, kF ) and the angle of incidence φ(ky) ∈ (0, π2).

This fact is represented by writing the x-component of wavevector, kx as well as φ both as

function of ky in Eq.(36). We take a finite beam with a gaussian envelope such that

f(ky − k) = exp[−(ky − k)2

2∆2k

] (37)

When ∆k � kF , we can approximate the ky-dependent terms by a Taylor expansion around

k and retaining only the first order term to get

φ(ky) ≈ φ(k) +∂φ

∂ky|k(ky − k); kx(ky) ≈ kx(k) +

∂kx∂ky|k(ky − k) (38)

Substituting in Eq.(36) and integrating, we obtain

Ψin =√

2π∆2kei[ky+kx(k)x]

e−∆2k

2[y−yin+ ]2

e−∆2k

2[y−yin− ]2eiφ(k)

, (39)

36

where

yin+ = −k′x(k)x, yin− = −k′x(k)x− φ′(k) (40)

Here the primed (′) quantities denote derivative taken wrt ky. Thus, upper and lower

components of the spinorial wave function are localized at separate points along the y-axis.

The reflected wavepacket can also be written in an analogous way by making the trans-

formation kx to −kx and φ to π − φ as well as multiplying the reflection amplitude

r(ky) = |r(ky)|eiφr(ky). The reflected wave is then

Ψr(x, y) =

∫ ∞−∞

dkyf(ky − k)eikyy−ikx(ky)xr(ky)

1

−se−iφ(ky)

(41)

Here again, s = sgn(EF ). The spatial profile of the reflected wave can be again obtained

by first expanding all ky dependent quantities around k and retaining only the first order

terms and then integrating in Eq.(41). This leads to

Ψr =√

2π∆2kei[ky−kx(k)x]|r(k)|

e−∆2k

2[y−yr+]2

−se−∆2k

2[y−yr−]2e−i[φ(k)−φ′r(k)]

(42)

Here, yr+ and yr− are given by

yr+ = −φ′r(k) + k′x(k)x, yr− = −φ′r(k) + k′x(k)x+ φ′(k) (43)

The above expression shows that the upper as well as lower components get shifted because

of the phase factor. The GH shifts of the upper and lower components are respectively given

by

σ+ = yr+ − yin+ = −φ′r(k) + 2k′x(k)x

σ− = yr− − yin− = 2φ′(k)− φ′r(k) + 2k′x(k)x (44)

Thus, the average shift for an MVP or EMVP barrier is

σ =1

2(σ+ + σ−) = φ′(k)− φ′r(k) + 2k′x(k)x (45)

The situation is depicted schematically in Fig. 15. The last term in the above expression is

a coordinate dependent quantity and will get an equal and opposite contribution from the

−φ′r(k) term. The resultant σ will thus be independent of the choice of the coordinate of

the interface from which TIR will take place.

37

FIG. 15: GH shift for an EMVP barrier. The solid line corresponds to the upper component of the

pseudospinor and the dotted line to the lower component. The GH shift can be either (a) positive

or (b) negative.

1. GH shift at p-n interface

To calculate the GH shift occuring at the p-n interface, we first require to calculate the

gradient of the phase of the refletion coefficient, the negative of which gives the GH shift.

For this we match the wavefunctions at the interface (x = −d) of the barrier such that

the propagating wavefunction on the left of the interface should match with the evanescent

wavefunction at the right of the interface.

ψGH1 =

eikxx + re−ikxx if x < −d

a′e−κ(x+d) if x > −d(46)

ψGH2 =

s[ei(kxx+φ) − re−i(kxx+φ)] if x < −d

−iγs′a′e−κ(x+d) if x > −d(47)

Here s = sgn(EF ), s′ = sgn(EF − V ), and

γ =~vF (κ+ ky)

E − V, k2

y − κ2 =

(E − V~vF

)2

TIR at p-n interface will take place when κ2 = E2F = (EF −V )2 > 0, for which the incidence

angle lies in the range φ > φc = sin−1(

VEF− 1)

. The continuity of the wave function at

x = −d gives the reflection coefficient as

r = e−ikxD[ieiφ − ss′γie−iφ + ss′γ

] = exp(−ikxD) exp(2iδ)

with tan δ = tanφ+ ss′γ secφ (48)

38

FIG. 16: Dependence on the angle of incidence φ of the GH shift σ, for U0/EF = 1.5 (dashed

curve, p-n interface) and for U0/EF = 0.4 (dotted curve, n-n interface).

Clearly, the phase of the reflection coefficient is φr = −kxD+ 2δ. The GH shift can now be

written as:

σ = φ′(k)− 2δ′(k), where δ = tan−1(tanφ+ ss′γ secφ) (49)

Using this expression, one can calculate the GH shift at p-n interface.

σ =sin2 φ+ 1− V/EF

κ sinφ cosφ(50)

This is the same expression as the one obtained in [99]. Since κ > 0 for decaying evanescent

solutions at the right of the p-n interface, for positive incidence angles the negative GH shift

at the p-n interface (EF < V ) will occur when sin2 φ− (V/EF − 1) < 0. Thus, for angles of

incidence φc < φ < φ∗ = sin−1√

sinφc, GH shift occurs in the backward direction. This is

explicitly shown in fig.16.

2. GH shift for MVP/EMVP barriers

For pure MVP barrier, TIR will take place when |sin |θ|| = sin |φ| + sgn(φ) 1kF lB

> 1.

This will happen when sin |θ| > 1 for 0 < φ ≤ π2

and when sin |θ| < −1 for −π2≤ φ < 0. In

the latter case, this requires the wave vector to be negatively refracted at sufficiently high

magnetic field before TIR occurs. This is already explained in section IV A in detail. The

GH shift can then be obtained with the expression as in eq.49 with only

γ =κ+ (ky + 1

lB)

kF,

(ky +

1

lB

)2

− κ2 =

(EF~vF

)2

39

FIG. 17: (color online) GH shift for an EMVP barrier at (a) 0.1T and (b) 3T. The y-axis corre-

sponds to the incident angle φ. The green regions have propagating solutions with no shift.

An EMVP barrier is essentially a p-n junction placed in a MVP barrier. For an EMVP

barrier, TIR occurs for electrons incident from both sides of the surface normal but at

different critical angles. For a given φ, it is possible to change V adiabatically and get the

electron wave reflected over a range of V satisfying |sin |θ|| > 1. We can keep φ fixed and

increase V . At V = Vc1 , TIR occurs when sin |θ| = 1 giving

Vc1E

= 1− [sin |φ|+ sgn(φ)1

kF lB]

Upon further increasing V , the electron wave remains totally reflected till V reaches the

second critical value Vc2 such that sin |θ| = −1 giving

Vc2E

= 1 + [sin |φ|+ sgn(φ)1

kF lB]

TIR occurs in the range V ∈ [Vc1 , Vc2 ]. In Fig. 17 is plotted the GH shift over the entire

range of V and φ. At all other regions in the φ− V plane the GH shift is set to 0 (green).

The explicit expression for the GH shift for an EMVP barrier in dimensionless form is

σkF =1− 2kF

k′Fcos2 δ(1 + tanφ tan δ + ss′ γ

2+1γ2−1

)

cosφ(51)

The γ2+1γ2−1

term in the numerator which is cothα or sec β diverges at the critical angle for TIR

and leads to a divergent GH shift. This is because just at the critical angle the wavevector

lies in the interface of the two regions. As a result in Fig. 17, the border TIR region of finite

GH shift shows the highest GH shift.

40

The upper and lower parts of the curve are again separated by the line φs = sin−1(− 1kF lB

). The left and right boundaries correspond to V < EF and V > EF . In the lower part, the

GH shift is mostly negative with the left and right boundaries having positive and negative

refraction respectively.

Comparing Eq.(51) with Eq. (50), we see that σ(φ) 6= −σ(−φ). This is due to the non-

specular nature of electron refraction at an EMVP barrier. This is an important difference

in the quantum GH effect that occurs upon TIR by an EMVP barrier as compared to TIR

by a purely electrostatic barrier for which |σ(φ)| = |σ(−φ)| as can be seen from Eq.50.

3. Lateral shift in the transmitted wave

As explained in the previous section, Goos-Hanchen shift is analogous to the phenomenon

of the lateral shift of the light beam total internally reflected from dielectric surface. Chen

et al. [97, 98] investigated the lateral shifts for Dirac fermions in transmission through

monolayer graphene barrier, based on tunable transmission gap. This shift has same physical

origin that is due to the beam reshaping since each plane wave component undergoes different

phase shift, however it has nothing to do with the evanescent waves which plays an all

important role in the lateral shift of total internally reflected wavefunction. For this reason

this shift can be termed as Goos-Hanchen like (GHL) shift and can be considered as an

electronic analogue of the lateral shifts of light beam transmitted through a metamaterial

slab.

According to stationary phase approximation [100] and on the same lines as explained

for GH shift for the reflected beam in th previous section, the GHL shifts of the transmitted

beam through a barrier in graphene can be obtained as [97, 98] σt = − δφtδky

∣∣∣ky0

where φt refers

to the phase of the transmission amplitude and the subscript corresponds to the wavevector

at central incidence angle. Using (11), and assuming the form t = |t|eiφt , the transmission

phase φt is obtained as

φt = tan−1[(ss′sec θ secφ− tan θ tanφ

)tan qxd

]so that

σt = −δφtδky

= d tanφ0 · T ·[(

2 +k2

0

k2x

+k2

0

q2x

)sin 2qxd

2qxd− k2

0

q2x

](52)

, where k20 = k2

y0 − ss′kFk

′F .

41

FIG. 18: Transmission gap (a) and GHL Shift (b) as a function of incident energy, E at φ = 300,

U = 62meV , d = 100nm.

Since the product ss′appears in the GHL shift expression so the analysis of the expression

can be classified into ss′

= −1 (Klein tunneling regime) and ss′

= +1 (classical motion

regime).

In the regime when ss′

= −1, k20 = kFk

′F + k2

y0 > 0 (for φ > φc and for all values of

incident energy, E). Then in accordance with (52), the maximum absolute value GHL shift

corresponds to qxd = nπ (These wavevector also corresponds to transmission resonances).

Hence

σtqxd=nπ = − k20

q2x0

d tanφ0

Thus in Klein tunneling regime, the GHL shifts are negative shifts when φ < φc.

In the other regime when ss′= 1, k2

0 = k2y0 − ss

′kFk

′F = −k′20 < 0 (for φ < φc and for all

values of incident energy, E). Then in accordance with (52), the maximum absolute value

of GHL shift corresponds to qxd = nπ (This wavevector also corresponds to transmission

resonances). Hence

σtqxd=nπ =k′20

q2x0

d tanφ0

42

FIG. 19: (Color online) (a) Contour plot of the transmission through a single barrier Vs its height

V and wave vector ky for D=100 nm. (a) B = 0 (b) B = 0.1T .

Thus in classical regime, the GHL shifts are always positive.

Thus we see that the GHL shifts in transmission can be negative or positive, and can be

enhanced by the transmission resonances when the incidence angle is less than the critical

angle for total reflection. This is clearly depicted in Fig.18.

D. Fabry-Perot Interference in Graphene Heterojunctions: Effect of magnetic

field

Another important optical analogue that can be studied in the ballistic transport regime

of graphene is the formation of Fabry-Perot resonances. Study of such resonances is the linear

and non-linear transport regime for non relativistic electrons for devices such as resonant

tunnelling diodes is well known for a long time particularly after the pioneering work by

[1, 101]. A excellent review on the experimental progress on such resonances in graphene

based heterostructures in the ballistic transport regime can be found in [15].

43

Below we revisit the analysis of potential barrier as a Fabry-Perot interferometer [41, 92]

and then we analyse that how the FP resonances get effected in the presence of magnetic

field in graphene.

The two edges of the square potential barrier acts as the two interfaces of FP interfer-

ometer [92], as shown in Fig. 20. As an electron wave is incident on the potential barrier at

an angle φ, it splits into the transmitted wave and reflected wave at the left edge (x = −d).

The transmitted wave propagate further to suffer another reflection(transmission) at the

right edge (x = +d). In this way it gets multiply reflected between the two edges of the

barrier. The difference in the optical paths between the transmitted wave t1 and t2 which

suffer consecutive reflections at the right interface is given as

∆L = 2(1− V/E)(BC + CD)−BP

=

(1− V

E

)2D

cos θ− 2D tan θ sinφ

=

(1− V

E

)2D

cos θ− 2D

cos θ

(1− V

E

)sin2 θ

=

(1− V

E

)2D cos θ using sinφ =

(1− V

E

)sin θ

This means that the net phase difference between t1 and t2 is δ = kx∆L = 2qxD.

Again we introduce r and t to be the reflection and transmission coefficient for the

potential step outside the barrier, and r′ and t′ the corresponding coefficients for inside the

barrier. In analogy with the optical waves, net transmission through the barrier is obtained

as

ttot = tt′ + tt′r′2eiδ + . . .+ tt′r′

2(n−1)ei(n−1)δ = tt′/[1− r′2eiδ]

.

Then, the total transmission probability T = t∗totttot is obtained as

T = 1/

[1 +

(4|r′|2

(1− |r′|2)2

)sin2 δ/2

](53)

Proceeding with the calculation given in section 2, the reflection and transmission coef-

ficients r, r′, t and t′ can be obtained by matching the wavefunction at the interface of the

44

potential step.

ψx<0 =

eikxx + re−ikxx

s[eikxx+iφ − re−ikxx−iφ]

ψx>0 =

t′eiqxx

s′t′eiqxx+iθ

Matching conditions at x = 0 gives

t = t′ =2 cosφ

s′/seiθ + e−iφ

r = r′ =1− s′/s cos(φ− θ)1 + s′/s cos(φ− θ)

(54)

Substituting 54 in 53 we obtain the total transmission probability of a single barrier.

T =cos2 θ cos2 φ

(cos qxD cos θ cosφ)2 + sin2 qxD(1− ss′ sin θ sinφ)2

Clearly the transmission resonances which are also the Fabry perot resonances occur at

qxD = nπ. The energies at which these resonance occur are obtained as:

E = V ± ~vF [n2π2/D2 + k2y]

1/2 (55)

Also T is equal to 1 at normal incidence φ = 0.

To show the effect of magnetic field which is already mentioned in sectionII B explicitly,

we consider a potential barrier of height V with a commensurate ⊥ magnetic field B [92]

V (x) = V, B = BΘ(d2 − x2)z |x| < d (56)

With the similar analysis as in section III B, the carriers in the presence of such potential

form, obey the Schrodinger like equation of the form:

[− ∂2

∂x2+ (kylB + x)2 + 1

]ψ1,2 = (ε− v)2 ψ1,2. |x| < d

Upon solving, the solutions can be obtained as:

ψ1(x) = aD(ε−v)2/2−1(√

2(x+ kylB)) + bD(ε−v)2/2−1(−√

2(x+ kylB))

ψ2(x) = ai√

2

(ε− v)D(ε−v)2/2(

√2(x+ kylB)) + b

−i√

2

(ε− v)D(ε−v)2/2(−

√2(x+ kylB))

45

FIG. 20: Electron transmitting through a potential barrier in (a) absence (b) presence of magnetic

field.

The solutions in the outside regions namely |x| > d remain same as in section III B. By

using the continuity condition to match the wavefunctions at the boundaries of the barrier we

numerically evaluate the transmission as plotted in fig.19. Since in the presence of magnetic

field the y-component of the momentum varies inside the barrier as ky → ky − eBx/~, so

the for the incidence angles at the two interfaces 1 and 2 to be of equal sign, ky(x1) and

ky(x2) should have opposite signs ie.

ky − eBx1/~ = ky + eBd/~ < 0

ky − eBx2/~ = ky − eB(d)/~ < 0

This gives the following condition on the y-component of momentum:

− eBd/~ < ky < eBd/~

A fringe contrast from color minimum of transmission to the color maximum of the trans-

mission on the two sides of the lines at ky = ±eBd can be clearly seen in the Fig.19 which

occurs as a consequence of the half a period shift in the transmission resonances. Occurance

of such phase shift for charge carriers in graphene was originally studied for a harmonic

potential in a uniform magnetic field by Shytov et al. [41] and used as an experimental

signature of Klein tunnelling [34]. This has already been discussed in section II B. Trans-

mission and conductance of massless Dirac fermions in the presence of multiple, disordered

short range scatterers can also be understood in terms of resonant transport through double

barrier structure. This has been discussed in detail in Ref. [84].

46

V. PERIODIC LATTICE OF MVP BARRIERS

Electron transport in graphene in presence of superlattices formed with electrostatic

potential or magnetic barriers lead to large a body work in recent times[19, 21, 102–104].

They lead to interesting physics and potent with the possibility of creating new devices in the

ballistic transport regime since they significantly alter the band structure. On other hand

they resemble closely the phenomena of light propagation through medium with periodically

modulated refractive index/dielectric constant which has been studied in detail in optics

literature and also led to many important applications [3, 105]. In the following section we

discuss this type of transport and their consequences. We shall also discuss the emergence

of additional dirac points in these modified band structure and the resulting collimation of

transport electrons using such superlattice structure.

A. Bandstructure modification

In the previous section we have shown how MVP/EMVP barrier affects the ballistic

transport in a profound way. Thus it will be interesting to find out how a periodic arrange-

ment of these barriers will effect the transport, what is the resulting band structure etc.. As

mentioned earlier, it has a direct optical analogue, namely with the electromagnetic propa-

gation in periodic stratified media following the pioneering work by Yariv and collaborators

[3]. It also provides us relativistic version of ordinary Kronig-Penny model and its magnetic

counterpart.

Practically there will only be a finite number of barriers present in the system and the

lattice translational symmetry will break down at the boundaries. However to simplify

the analysis, we assume that the unit structure- a double EMVP barrier structure can

be repeated infinitely along the x-axis. The magnetic field creating such a barrier B =

Bz(x)z = BlB[δ(x+d)+ δ(x−d)−2Bδ(x)]z together with a superimposed splitgate voltage

geometry, hence forth will be called DEMVP barrier. Since changing or reversing V can

locally convert a charge-neutral region into a p-n or n-p junction, such a periodic barrier

can also be thought of as a semiconductor heterostructure. It should be emphasized that

this can lead to new device structures due to combined effect of the highly inhomogeneous

and periodic magnetic fields and controllable voltages. The vector and scalar potentials are

47

characterized by

V (x) = −V ;Ay(x) = B`B,−d < x < 0 : region I

V (x) = V ;Ay(x) = −B`B, 0 < x < d : region II (57)

The schematic diagram in Fig. 9(b) shows how such DEMVP barriers can be constructed

by placing metal electrodes and insulating layers between PMA materials with alternating

magnetizations and the graphene sheet.

We consider each unit cell of size D = 2d for the MVP as well as for the electrostatic

potential barriers. Thus, the n-th cell is given by the region between (n− 1)D and nD. In

the α-th part of a unit cell, the wavefunction is

ψ1 = aαneiqnαx(x−nD) + bαne

−iqnαx(x−nD) (58)

ψ2 = sαn[aαne

i[qnαx(x−nD)+θα] − bαne−i[qnαx(x−nD)+θα]

]Here,

α = 1, 2; a1n = an, b

1n = bn, a2

n = cn, b2n = dn;

s1,2n = s1,2; s1,2 = sgn(E ± V ) qn1x,2x = q1,2 (59)

The wavevectors q(1,2) are given as

q21 + [ky +

1

lB]2 = [

EF + V

~vF]2

q22 + [ky −

1

lB]2 = [

EF − V~vF

]2

The exponential factor e−nD reveals the existence of lattice translational symmetry, which

is not present for the isolated EMVP and DEMVP barriers.

The continuity of the wavefunction at the first interface at x = (n− 1)D gives 1 1

s2eiθ2 −s2e

−iθ2

cn−1

dn−1

=

e−iq1D bneiq1D

s1ei[q1D−θ1] −s1e

i[q1D−θ1]]

anbn

(60)

Similarly, the continuity at the second interface at x = (n− 1)D + d gives e−iq1D2 eiq1

D2

s1e−i[q1D2 −θ1 ] −s1e

i[q1D2−θ1]]

anbn

=

e−iq2D2 eiq1

D2

s2[e−i[q2D2−θ2]] −s2e

i[q2D2−θ2]]

cndn

(61)

48

FIG. 21: Band structure at some characteristic values of V with B-field strength strengths (a) 3

Tesla, and (b) 0.1 Tesla.

Introducing a shorthand notations like,

MA =

e−ikx 0

0 eikx

; MB1,2 =

e−iq1x,2xd 0

0 eiq1x,2xd

Ms,s1,2 =

1 0

0 s, s1,2

; Mθ1,2 =

1 1

eiθ1,2 −e−iθ1,2

Now Eqns.(60) and (61) can be rewritten as

Ms2,n−1Mθ2

cn−1

dn−1

= Ms1,nMθ1MB1

2

anbn

Ms1,nMθ1MB1

anbn

= Ms2,nMθ2MB2

cndn

(62)

The above two matrix equations can be combined as

cn−1

dn−1

= M−1θ2M−1

s2Ms1Mθ1MB1M

−1θ1M−1

s1Ms2Mθ2MB2

cndn

=

K11 K12

K21 K22

cndn

(63)

According to Bloch theorem, cn−1

dn−1

= e−iKD

cndn

(64)

49

The matrix Kmat =

K11 K12

K21 K22

is unimodular. From Eqs.(63) and (64) we obtain the

eigenvalue equation K11 K12

K21 K22

cndn

= e−iKD

cn−1

dn−1

(65)

where K is the Bloch momentum. The complex conjugate eigenvalues λ are given by

det|Kmat − λI| = 0.This implies λ1 + λ2 = exp(−iKD) + exp(iKD), which finally gives

K(φ,B) =1

2dcos−1[

1

2Tr(Kmat)] (66)

Writing in terms of the wavevectors q1, q2 and the angles θ1, θ2, the above eigenvalue

condition reads

cosKD = cos q1d cos q2d+ sin q1d sin q2d×[tan θ1 tan θ2 −

s1s2

cos θ1 cos θ2

](67)

This equation provides the band structure for a periodic DEMVP barrier superlattice in

general for any V 6= E. By substituting V = 0 in the above expression, the band structure

for a periodic DMVP barrier superlattice can be obtained. This is an extension of the Kronig-

Penney (KP) model to two-dimensional massless Dirac fermions. Thus, it is interesting to

compare the DEMVP band structure with other variants of the KP model. The original KP

model describes Bloch waves in a one-dimensional periodic potential [4]. Several authors

have also studied the relativistic KP model [106–109], where the motion considered is strictly

one-dimensional. The non-relativistic KP model in periodic structures created by MVP

barriers has also been studied [20, 110]. For graphene charge carriers, such problems have also

been studied for different types of periodic magnetic [19, 21, 103, 104] as well as electrostatic

[102] barriers. Magnetotransport studies in presence of periodic barriers [111] where the

periodic modulation is sinusoidal was carried out for monolayer graphene and it was found

that such periodic modulations Landau levels into bands which oscillate with the strength

of the magnetic field.

In Fig. 21 is plotted the band structure of corresponding to the periodic DEMVP barriers

as a function of magnetic field B and voltage qVEF

. Conducting regions are seen over a wide

range of E with forbidden regions in between. At 0.1T, the region near V = 0 is mostly

conducting for different values of energy where a forbidden region starts opening up both to

the left and right of the V axis. The gaps are much wider than their counterparts for pure

50

FIG. 22: The group velocity vg (in units of vF ) at the neutrality point as a function of d/lB.

MVP barriers. Close to qV → EF , an extended forbidden zone appears. To understand

more explicitly, the band structure is also plotted at qVEF

= 0.7 close to the singular point.

Here, the conducting regions are intervened by large patches of forbidden zones. In

comparison, at qVEF

= −0.7, the behaviour is completely changed and the forbidden zone over

the same range of kylB shrinks considerably (left column). For higher V where qV > EF ,

the system is conducting at almost all E. At 3T and near zero V , a forbidden region opens

up at various values of the energy and is much larger than at 0.1T. The gapped regions

to the left and right of zero V at 3T are located in a pronounced asymmetric manner as

compared to when B = 0.1T. This asymmetry in the band structure as a function of V

as well as the opening up of large forbidden zones for certain values of V differentiates the

transport through EMVP barriers from MVP barriers and thus, it provides more flexibility

to tune such transport.

As shown by various studies, carriers in graphene superlattices exhibit several interesting

features [112] [113] [91] that result from the particular electronic SL band structure. Below

we briefly analyse the bandstructure of periodic DMVP barrier superlattice.

1. Dispersion in the vicinity of neutrality point

We begin with analysing the dispersion relation in the vicinity of the neutrality point i.e..

close to zero energy. By rewriting the Eq.67 in the form of f(E, kx, ky) = 0, then expanding

the f -function to the lowest order in E and ky and then in kx in the vicinity of the neutrality

51

FIG. 23: The circles emphasize the finite energy Dirac points at the values corresponding to Eq.

69, for B = 0.5T , and d = 100nm.

point (E = 0, ky = kx = 0), we obtain the dispersion relation:

E = ±~vFv0

√k2x + k2

y, with v0 =d/lB

sinh(d/lB)(68)

Clearly, the above dispersion relation is isotropic (∂E/∂kx = ∂E/∂ky) with zero gap between

the valence band and conduction band. Moreover the group velocity vg = v0vF depends on

the width of the unit DMVP barrier as well as on the B−field strength. The group velocity

is plotted in Fig. 22, which clearly shows that vg is always smaller than the Fermi velocity

and it monotonously decreases for increasing d/lB, which is implied from the single MVP

barrier analysis as well.

2. finite energy Dirac points

Apart from the neutrality point described in Eq. 68, many other degeneracy points can

be seen in Fig.23 where the dispersion have a double cone like structure. The locations of

these points occuring along the ky = 0 direction can be obtained analytically as

En = ±~vF√

(1/lB)2 + (nπ/d)2 (69)

52

FIG. 24: Plots of the dispersion relation at n = 1 (top row) and n = 3 (bottom row) for B = 0.1T

(left column) and B = 1T (right column), d = 50nm.

Again by expnading the f -function in the vicinity of these points En(kx = 0, ky = 0), the

dispersion relation is obtained as

E = En ± ~vF√v2nxk

2x + v2

nyk2y, n = 1, 2, ... (70)

with the group velocity components given by

vnx =(nπ)2

(d/lB)2 + (nπ)2

vny =(d/lB)2

(d/lB)2 + (nπ)2

Clearly the dispersion relation is linear in kx and ky, which shows that in the presence

of DMVP barrier superlattice, there are infinite number of degeneracy points close to which

the dispersion has a double cone shape which is asymmetric in kx and ky direction. These

points are termed as ”Extra Dirac points” in literature[114][115].

In contrast to the dispersion relation at the neutrality point which is isotropic in nature,

the dispersion at these finite energy extra Dirac points is anisotropic (∂E/∂kx 6= ∂E/∂ky).

As determined by Eq.70, we plot in Fig.24, the dispersion relation at n = 1, 3 for different

B−field strengths, B = 0.1, 1T . From this we infer a collimation along ky direction ie.

vy ∝ ∂E/∂ky ≈ vF and vx ≈ 0 as the contours become almost independent of kx. This is

similar to that found for a superlattice of electrostatic potential barriers [91].

53

VI. THE TRANSFORMATION OF MAGNETIC TO ELECTRIC FIELD FOR

DIRAC FERMIONS IN GRAPHENE: A CASE FOR RELATIVISTIC INVARI-

ANCE

For the electron transport in presence of a transverse uniform magnetic field it was shown

by Lukose et al. [116] that the Landau levels gets modified when the system is subjected to

an electric field, finally leading to the Landau level collapse through the renormalization of

the cyclotron frequency. A similar rescaling of the magnetic field strength took place when

a suitable gate voltage is applied for the magnetic barrier problem as well [21] as discussed

in section IV B. To summarize one can change the strength of a magnetic field by suitably

applying a get voltage. A more detailed analysis done later on [117] how the relativistic

invariance of the effective equation that governs the charge carriers in graphene leads to

this interconvertibility of electric and magnetic field. In this section we shall describe this

approach.

According to Special theory of Relativity, the laws of physics must be formed in a form

so that it do not allow one to distinguish between the frames of reference which are moving

with a constant relative velocity. The transformation between such frames are described by

Lorentz transfromations. Like Maxwell’s equations, Dirac equation is also Lorentz covariant.

So, for the charge carriers in graphene obeying ”Dirac-like” equation, the Lorentz covari-

ant structure of Hamlitonian can be used to find an analogy with special relativity. The

analogy extends to problems involving coupling of electrons to external electric or magnetic

fields. It is known from from special Relativity that in the presence of electromagnetic fields

there are two invariants; namely E2 − B2 and E.B which remain unchanged in transition

from one inertial frame to another [118]. This means, by means of Lorentz transformation

we can always give E and B any arbitrary values, subject to only condition that E2 − B2

and E.B have fixed values. Particularly, if E.B = 0, then we can always find a reference

frame in which E = 0 or B = 0 (according as E2 − B2 < or > 0), ie. the field is purely

electric or purely magnetic. In the following discussion we’ll first find the transformation

laws for the graphene charge carriers and then we analyse the consequences of application

of Lorentz boost [116, 117, 119]. For this, it will be convenient to write graphene equation

in Lorentz - covariant form [119]:

γµ (pµ − aµ) Ψ = 0 (71)

54

where the 2 × 2 γµ matrices satisfy the anticommutation relation [γµ, γν ]+ = gµν ; gµν is

the metric -tensor. We can explitly write the gamma-matrices as: γ0 = σz, γ1 = −iσy,

γ2 = −iσx, where σx, σy, σz are Pauli matrices. Here we use the space-time notation for

coordinates as xµ = (vF t, x1, x2), momenta pµ = ~(iv−1F ∂t,−i∂x1,−i∂x2), and external field

aµ = (a0, a1, a2) =(− VvF,− eAx

c,− eAy

c

).

The above Eq.71 is invariant under Lorentz transformation. The transformation laws can

be obtained on the same lines as for the general (3 + 1) Dirac equation. The space-time

coordinates, momenta and fields transform as:

xµ′

= Λµ′

µ xµ, pµ′ = Λµ′

µ pµ, aµ′ = Λµ′

µ aµ (72)

Ψ′ = S(Λ)ψ

where S(Λ) = exp(

18ωµν [γ

µ, γν ])

for Λ = exp(ω).

We consider the magnetic field modulations where the field strengths vary in x direction

and are constant in y-direction and scalar potential V = V(x), then a1 = 0 in Eq.71. Writing

the solutions as Ψ(x, y, t) = eikyye−iEt/~, we obtain

Eψ(x) =[−i~vFσx∂x + vFσy

(~ky +

e

cA(x)

)+ V (x)

]ψ(x) (73)

Now we apply two set of transformations [117] (A) a lorentz boost in y direction with

rapidity iπ/2, followed by (B) a reflection about the y-axis. These two transformations

together make a complex Lorentz boost as det(ΛA)det(ΛB) = 1. With this complex Lorentz

boost applied the energy and momentum transforms as :

E = −i~vFk′y

ky = iE ′/~vF

And transformed Eq. 73 becomes

E ′ψ′(x) =[−i~vFσx∂x + σy

(~vFk′y − iV (x)

)+evFciA(x)

]ψ′(x) (74)

On comparing 73 and 74 it can be clearly seen that it is the imaginary scalar potential

which is now coupling with the y-component of momentum and the role of scalar potential

is now played by the external vector potential (again imaginary).

This section concludes our reviewing various aspects of ballistic electron transport in

monolayer graphene through magnetic barriers, tuned by gate voltage and the associated

optical analogy. In the subsequent section we shall discuss the electron transport in bilayer

graphene in presence of such magnetic barriers.

55

VII. MAGNETIC BARRIERS IN BILAYER GRAPHENE

A. Transmission through magnetic barriers

In this section, we consider the effect of such inhomogenous magnetic fields for the case of

unbiased bilayer graphene (BLG). Bilayer graphene is modelled as two coupled hexagonal

lattices including inequivalent sites a, b and c, d in the bottom and top layers respectively.

The present work discusses bilayer graphene in bernal stacking. In such an arrangement, the

bottom and top layers are shifted with respect to each other in such a manner that c sites are

exactly above the a sites and b sites are exactly in the middle of the hexagons of the bottom

layer. In addition to intralayer interactions, there are interlayer interactions present in the

system. The band structure is derived after taking into account two possible ways of A C

hopping [120]: via the dimer state or due to weak but direct a− c coupling. There are other

weaker tunnelling processes [26] which are neglected in this tight-binding approximation

which we consider. The tight-binding Hamiltonian [120] derived in this way spans over the

basis states that are located at a, b, c, d. The effective low energy Hamiltonian near the Fermi

level [120] obtained within the k · p approximation from this tight binding Hamiltonian in

this case is different from the corresponding one given in Eq.(5) for monolayer graphene.

The effective 2× 2 Hamiltonian is given as

Heff = − ~2

2m

0 (kx − iky)2

(kx − iky)2 0

The solutions of these equations are massive Dirac fermions since they have a quadratic

dispersion and non-zero effective mass m. The applicability of this model Hamiltonian to

study transport through a scalar potential was discussed in detail in [122]. In the presence

of scalar potential V (x), under normal incidence ky = 0, pseudospin σx is still conserved,

but the Hamiltonian is like that of a Schrodinger particle; i.e., Heff = − ~2

2mk2xσx + V (x)I.

The probability amplitude of corresponding charge carriers exponentially decays like usual

Schrodinger particles inside the barrier leading to negligible transmission [14, 38, 121]. Draw-

ing an analogy with optical transmission we call this phenomenon Klein reflection.

Below we calculate the effect of magnetic barrier for the case of bilayer graphene (BLG).

The electronic wavefunction depends on the spatial profile of the vector potential and the

boundary conditions when an electron is incident from a region with B = 0 to a region where

56

B 6= 0. We show this by explicitly calculating transmission by a transfer matrix approach

[123], [124]. Chiral charge carriers in BLG obey:

i~∂Ψ(x, y)

∂t= HΨ, H =

V1 Π εt 0

Π+ V1 0 0

εt 0 V2 Π+

0 0 Π V2

(75)

Here, they are described by a 4-component spinor Ψ(x, y) = (Ψa Ψb Ψc Ψd)T and a 4 × 4

Hamiltonian in the presence of a magnetic barrier and an electrostatic potential. Π =

vF [px + i(py + eA/c)] with vF = 106 m/s and εt is the tunnel coupling between the two

layers. V1 and V2 are the potentials at the two layers. The field profile taken here is

B = BΘ(x2 − d2)z. As pointed out earlier that such a sharp profile is routinely created

in magnetic recording media where bit lengths can be varied over 10 − 100nm range using

suitable domain engineering [50, 125]. Since this length scale is much larger than the A-A

lattice constant of 0.246nm in graphene but also much smaller than the Fermi wavelength

λF = 50nm (at EF = 17meV) of the incident electrons in bilayer graphene, the inhomogenous

field pattern can be well approximated by the sharp edged profile proposed here. For bilayer

graphene also, such sharp fields produced either by a patterned media layer or by a close-

packed nanowire array can be easily integrated in the SiO2 underlayer present in typical

top- and bottom-gated graphene devices for Klein tunnelling studies [34]. The fields and

voltages needed are within range of values already being used.

We are interested in the stationary state solutions of the form Ψ = ψ(x, y)e−iEt/~. Since

[H, py] = 0, we can assume solutions like Ψα(x, y) = φα(x)eikyy, for α = a, b, c, d. Substitut-

ing this into Eq.75, and using lB =√

~ceB

and εB = ~vFlB

as units of length scale and energy

scale respectively, to define the dimensionless quantities x → xlB

, v1,2 → V1,2

εB, ε′t → εt

εBand

ε→ EεB

, we obtain the following set of coupled equations:

− i[d/dx− (ky + eAylB/~c)]φb + ε′tφc = (ε′ − δ)φa

−i[d/dx+ (ky + eAylB/~c)]φa = (ε′ − δ)φb

−i[d/dx+ (ky + eAylB/~c)]φd + ε′tφa = (ε′ + δ)φc

−i[d/dx− (ky + eAylB/~c)]φc = (ε′ + δ)φd (76)

57

FIG. 25: Transmission, T Vs V and qy for various fields. (a) Barrier, (b) Barrier+well. ∆ = 0,

d = 50nm, EF = 17meV.

Here

Ay =

sgn(x)Bd/2 if|x| > d2

Bx if|x| < d2

(77)

sgn(x) represents sign of x, ε′ = ε− v0 with v0 = (v1 + v2)/2 = V+

εBand δ = (v1− v2)/2 = ∆

εB.

The solutions can be explicitly obtained in the three regions x < −d2

(region I), −d2≤

x ≤ d2

(region II) and x > d2

(region III) as follows:

58

Region I(j = 1) and III(j = 3) are constant vector potential regions. We substitute

φα(x) = φα(0)eiqxx in Eq.76. With this, the dispersion relation becomes:

[−q(x)2 + (ε′ + δ)2][−q(x)2 + (ε′ − δ)2] = (ε′2 − δ2)t′

2(78)

Here, q(x)2 = q2x + q2

y with qy(x) = ky + sgn(x)π ΦΦ0

. Also, ΦΦ0

is the total magnetic flux

through an area dlB in units of the flux quantum Φ0 = hce

. The above Eq.78 leads to

propagating as well as evanescent wave solutions. The unbiased BLG (which is the case

considered here), corresponds to δ = 0, and v1,2 = v0 = 0 inregion I and III. Then the

complete wave function in region I and III can be given as:

Ψj(x, y) =M0(x)eikyy(aj bj cj dj)T (79)

M0(x) =1

ε′

ε′eiqxx ε′e−iqxx ε′e−κxx ε′eκxx

[qx − iqy(x)]eiqxx −[qx + iqy(x)]e−iqxx i[κx − qy(x)]e−κxx −i[κx + qy(x)]eκxx

−ε′eiqxx −ε′e−iqxx ε′e−κxx ε′eκxx

−[qx + iqy(x)]eiqxx [qx − iqy(x)]e−iqxx i[κx + qy(x)]e−κxx −i[κx − qy(x)]eκxx

For the wavefunction solutions in region II, where the magnetic field is finite, we introduce

(x+ ky) = z/√

2 and rewrite Eq.76 as:

− i√

2[d/dz − z/2]φb + ε′tφc = (ε′ − δ)φa

−i√

2[d/dz + z/2]φa = (ε′ − δ)φb

−i√

2[d/dz + z/2]φd + ε′tφa = (ε′ + δ)φc

−i√

2[d/dz − z/2]φc = (ε′ + δ)φd (80)

With this, we obtain two coupled equations in φa and φc which looks like((ε′ + δ)2 + 2(d2/dz2 − z2/4− 1/2)

)φc = (ε′ + δ)ε′tφa(

(ε′ − δ)2 + 2(d2/dz2 − z2/4 + 1/2))φa = (ε′ − δ)ε′tφc (81)

The above two equations can be solved to obtain a fourth order differential equation

which can be decomposed in two second order differential equations. For φa, we obtain :

(d2/dz2 − z2/4 + γ+/2)(d2/dz2 − z2/4 + γ−/2)φa = 0,

γ± = ε′2

+ δ2 ± [(1− 2δε′)2

+ (ε′2 − δ2)ε′t

2]1/2 (82)

59

The above equation admits the solutions which are of the form of parabolic cylindrical

functions [70]. For the case of unbiased BLG the complete wavefunction solution in region

II can be given in terms of a compact matrix as:

MB(x) =

Dp+(z) Dp−(z) Dp+(−z) Dp−(−z)

ε∗1p+Dp+−1(z) ε∗1p

−Dp−−1(z) ε1p+Dp+−1(−z) ε1p

−Dp−−1(−z)

ε+2 Dp+(z) ε−2 Dp−(z) ε+

2 Dp+(−z) ε−2 Dp−(−z)

ε1ε+2 Dp++1(z) ε1ε

−2 Dp−+1(z) ε∗1ε

+2 Dp++1(−z) ε∗1ε

−2 Dp−+1(−z)

Here D corresponds to parabolic cylindrical function of argument z and order p± =

(γ± − 1)/2. Also ε1 = i√

2ε′

and ε±2 = ε′

ε′t− 2p±

ε′tε′ , and γ± = ε′2 ±

√1 + ε′2ε

′2t . The current

density expression is obtained as jx = vFψ+

σx 0

0 σx

ψ. The transfer matrix through any

combination of a scalar and vector potential can now be written in terms of transfer matrices

MB(x) and M0(x) for regions with finite and zero magnetic field respectively. This can

then be used to find the transmission. The ratio of current density in region III and the

incident current density in region I gives the transmission probability as a function of the

angle of incidence φ′ = tan−1 qyqx

.

In region I we parametrize q = (qx, qy) in terms of incident energy, (ε > 0) and the angle

of incidence φ′ as (qx, ky − d2) = (

√εε′t + ε2 cosφ′,

√εε′t + ε2 sinφ′). In region III similar

parametrization can be done in terms of the angle of emergence θ′ to write (q′x, ky + d2) =

(√εε′t + ε2cos θ′,

√εε′t + ε2 sin θ′). The conservation of energy and constancy of the normal

component of momentum then ensure

sin θ′ =d√

εε′t + ε2+ sinφ′

This equation explains that beyond a certain critical angle the electron beam and its

cross-section (allowed φ values) is reduced with increasing barrier strength. This feature,

along with chiral symmetry breaking, dictates the nature of transmission as is subsequently

explained.

In figure 25 is plotted transmittance T for a magnetic barrier ((a), left column) and for

a barrier + well ((b), right column). In figure 1(a), the uppermost plot gives T for B = 0T ,

clearly showing the region of perfect reflection around normal incidence symmetrically placed

between two wings of resonant FabryPerot fringes. This is a generic feature of transmission

60

FIG. 26: G for different magnetic barrier strengths. (a) Barrier, (b) Barrier+well. ∆ = 0, d =

50nm, EF = 17meV.

through such barriers [41][123]. As the barrier strength increases to B = 0.2T , a transmission

region develops between these two wings due to the effect of the magnetic barrier and

the resulting transmission also becomes asymmetric. For higher fields, the disappearance

of perfect Klein reflection at normal incidence is seen even more clearly. We may then

conclude that in BLG, though the passage from electron to hole states at normal incidence

is usually forbidden due to pseudospin conservation, even a weak magnetic field lifts this

restriction. For a weak magnetic field, the coupling between electron and hole states is also

weak; however, for a case of resonant scattering one can expect total transmission [126] The

total angular range of transmission however shrinks in the presence of the magnetic barrier

since, beyond a certain incident angle, all electron waves suffer total internal reflection. In

figure 1(b) is plotted T through a barrier + well, consisting of two magnetic barriers equal

in magnitude and opposite in direction such that the total flux through the region vanishes.

Now, the incident and the transmitted wavevectors are parallel to each other. As a result,

the net rotation of the pseudospinor due to the inhomogeneous field vanishes and Klein

reflection at normal incidence is restored. The FabryPerot fringes bend due to asymmetric

transmission at a given V.

B. Conductance

We now study the effect of the above transmission on conductance at very low temper-

atures and for energies close to the Fermi energy in the linear transport regime. To a good

61

approximation, the dimensionless conductance can be written as [73]

G =

∫ π2

−π2

dφT (E, φ) cosφ (83)

Fig.26 plots conductance G as a function voltage. This shows that the gaps between the

conductance maxima and minima get reduced in the presence of a magnetic barrier. The

barrier reflects electrons incident upon it beyond a critical angle and the angular range of

transmission shrinks with increasing barrier strength. As a result, the absolute value of con-

ductance maxima comes down. Also the asymmetry, namely the way G changes to the right

and left side of such maxima, reduces in the presence of magnetic barrier. The behaviour

of the conductance is strongly dependent on transmission resonances. To explain this be-

haviour, therefore we shall now model these numerically computed transmission resonances

using a the Breit Wigner model [127]

t(ε′, ky) =Γ(ky)

Γ(ky) + i(ε′ − ε′∗(ky)(84)

Such a form is valid near the transmission resonance. The above model has been widely

used to model the transport due to resonant tunnelling in semiconductor structure [1].

Recently this model has been used to understand the cloaking phenomena in bilayer graphene

[128]. Here ε′∗(ky),Γ(ky) are respectively the momentum dependent resonant energy and the

resonance width.

Following the expression (78) in the absence of a magnetic field, the resonance energy

ε′∗(ky) and the resonance width Γ(ky) can be assumed to have a quadratic dependence on the

wavevector ky [128]. Thus ε′∗(ky) ≈ ε′∗(0)+αk2y and Γ(ky) = βk2

y. The quadratic dependence

of the resonant energy and the transmission width explains why the Fabry-Perot fringes of

transmission resonance depicted in the left column of Fig. 27 are of convex shape. The

asymmetry in the form of this fringes reflects themselves in the plot of the conductance

again depicted in the left lower corner of Fig. 27, as one goes from the conductance peak

to the left and right along the ε′ axis (effectively the voltage axis). In order to analyse the

conductance peak, we evaluate the quantity δG(ε′) = G(ε′) − G(ε′∗0), Here G(ε′∗0) repre-

sents the conductance value at the peak, and G(ε′) represents the conductance value in its

neighbourhood. Then,

δG(ε′) = G0

∫ (|t(ε′, ky)|2 − |t(ε′∗0, ky)|

2)dky

62

FIG. 27: Explanation for the conductance: The upper row consists of zoomed in portion of the

plot given in Fig. 25 and the lower row consists of the zoomed in portion of Fig. 26.

On substituting the quadratic dependencies of reseonance energy and resonance width in

the integral,

δG(ε′) =G0β

2

α2 + β2

∫ +kf

−kfdky

2(δε′)αk2y − (δε′)2[

(βk2y)

2 + (δε′ − αk2y)

2]

which upon solving gives:

δG(ε′) = −G0πcos[3ω

2+ π

4sgn(δε′)]

(1 + α2)3/2

√|δε′|β, α = α/β (85)

Clearly δG(ε′)δε′

shows the sqaure root sigularities at the conductance peak and the sgn(δε′)

term appearing in the expression introduces the asymmetry on the two sides of resonance

energy.

In the presence of a finite magnetic barrier, the small ky behaviour of such resonances

can be obtained by using a semiclassical argument. We replace ky by its covariant form,

namely ky + 2πBdhce

. This yields ε′∗(ky) ≈ ε′∗(0) +Ad2 where A is a constant. With increasing

strength of the magnetic barrier, ε′∗(0) is shifted by the amount of the flux inserted in

the barrier regime and this shift is independent of ky. The resonance width also follows a

similar behaviour. The right column of figure 3 depicts this situation. The field-induced

resonance leads to a central fringe symmetric about the voltage axis, unlike the upper and

63

FIG. 28: Selective blocking of modes in trilayer graphene. E and k are in arbitrary units. The

scalar potential V blocks massive modes while the massless mode is passed through. The massless

mode is weakly transmitted through the magnetic barrier B.

lower branches, which are asymmetric. This is again seen in the conductance plot as a right-

left asymmetry around the conductance peak, which is now reduced. The reduction of the

conductance peak can be attributed to the shrinking of the angular range of transmission due

to the magnetic barrier, whereas the rounding of the conductance peak and the consequent

reduction in the left and right asymmetry around this peak is an indicator of the reversal

of Klein reflection. Since conductance is a physically measurable quantity, this provides a

direct way to experimentally test our results.

C. Generalisation beyond bilayer

The above findings also extend to multilayer graphene. For N-layer graphene with Bernal

(ABAB · · · ) stacking The dispersion relation is given by [129]

E±r,p = t cos(rπ

N + 1)±

√v2F |p2|+ t2 cos2(

rπ

N + 1)

where r = 1, 2, 3, · · ·N corresponds to the N modes of the energy spectrum, vF is the Fermi

velocity. For odd N , r = N+12

is massless and it’s energy is given by

E±N+12,p

= ±vF |p| (86)

64

For all other N − 1 cases the dispersion relation takes the form E2 + 2Et cos( rπN+1

) = v2|p|2.

In the limit |E| << εt the low energy spectrum for the r-th massive mode is given as

Er,p =p2

2mr

, if εt cos(rπ

N + 1) < 0

= − p2

2mr

, if εt cos(rπ

N + 1) < 0

where we define mr =|εt cos( rπ

N+1)|

v2F

as the effective mass of the r-th massive mode. Such band

structure has been plotted in fig.28 for trilayer graphene. Now the transport electrons near

the Fermi level will be a mixture of eigenmodes, one of which is a massless Dirac mode

and the others are massive modes with different effective masses and also have a different

chirality. These arguments extend to N > 3 also. An extension of the semiclassical argument

presented earlier therefore suggests the following possibility in the case of such multilayer

graphene. When such a system is exposed to a scalar potential barrier, a normally incident

massless mode will undergo Klein tunnelling, whereas in the case of a massive mode there

will be exponentially decaying transmission. In contrast, a magnetic barrier will be highly

reflective for a massless mode but will transmit normally incident electrons for massive

modes. A typical situation has been depicted in fig.28. Since the low energy band structure

decomposes into one massless and N−1 massive modes for N odd and only N massive modes

for N even [129], our results qualitatively suggest that a magnetic barrier can be used to

selectively allow or partially filter out a mode.

In Fig.29 we compare the modified bandstructure obtained with a periodic arrangement

of DMVP in monolayer and bilayer graphene. One dimensional (1D) periodic modulation

of the chemical potential or the electric field perpendicular to the layers was addressed in

Ref. [130]. For a periodic stratified media of several DMVP barriers, forbidden region

appear in the bandstructure where the propagation of Bloch waves is not allowed. Th size

of these forbidden regions increase with increasing strength of magnetic field. A comparison

between the plots in Fig.29 (a),(c) and (b),(d) shows very clearly the conical band structure

of monolayer graphene and parabolic bands in bilayer graphene.

This issue of anti-Klein tunnelling effect in bilayer graphene p-n-p junction and the effect

of magnetic field on it, has also been addressed Ref. [128]. Here they describe it in terms

of the bound states being completely decoupled from outside continuum states, thus calling

it an electronic cloaking effect, and the barrier acting as ”cloak”. The effect is complete at

normal incidence, leading to ”Klein reflection” or ”anti-Klein tunneling”. This is explained

65

FIG. 29: The band structure modification of monolayer and bilayer graphene in presence of mag-

netic barrier. (a) and (c) plots the bandstructure of monolayer graphene in presence of periodic

arrangement of magnetic barriers with two different strengths where as (b) and (d) plots the

corresponding bandstructure of bilayer graphene.

as follows: At normal incidence the effective equation that describes the motion of charge

carriers in graphene , (as can be seen from Eq.75), can be analysed in the eigen basis of σx:(− ~2

2m

d2

dx2± [V − E]

)ψ±(x) = 0 (87)

Clearly, the solutions for ψ+ corresponds to the solutions outside the barrier. And the

solution for ψ− (for which the equation appears with a negative sign) are bound state

solutions, and corresponds to the solutions inside the barrier. Thus these two kinds of

solutions are completely decoupled leading to zero transmission despite of the states available

inside the barrier.

In section VII, our discussion on Klein reflection in bilayer graphene was based on the

analysis of Fabry-Perot like transmission resonances developed from inhomogenous magnetic

barriers in the presence of voltage. Here the oscillatory transmission due to Fabry-Perot like

arises due to interference of forward and backward moving propagating waves which undergo

multiple reflection between two interfaces. In addition to these FP like resonances a new

kind of transmission resonances have also been seen for the case of gapped bilayer graphene

66

[131]. In their recent work, R. Nandkishore et. al. proposed an entirely different approach to

realize oscillatory transmission which involves only forward propagating waves and a single

interface and thus is different from the FP resonances. They show that Zener transmission

in a p-n junction in gapped bilayer graphene exhibits common path interference taking

place under the tunnel barrier, leading to transmission that oscillates as a function of gate-

tunable bandgap.The origin of this oscillatory behaviour lies in the fact that in the presence

of p-n junction, bilayer graphene employs the propagating as well as evanescent wavevector

solutions. The simultaneous existence of these two- leads to common path interference which

gives rise to oscillatory transmission. These oscillations in transmission manifest themselves

through negative differential conductivity in the I−V characteristic. Interesting works were

also done on pseudo spinotronics in biased bilayer graphene [132, 133] which we shall not

discuss in this review. Other recent developments made on these lines include the work by

Campos et. al. [134], where they have realized giant conductance oscillations in ballistic

trilayer graphene Fabry-Perot interferometers, which result from phase coherent transport

through resonant bound states beneath an electrostatic barrier.

VIII. CONCLUSION

A large body of work has followed since the proposal of using inhomogeneous magnetic

fields in graphene and understanding the transport in an analogy with light beam propa-

gation in optics- termed as electron optics. The similarities between polarization states of

light and ballistic charge carriers in graphene has been investigated in detail in [135, 137].

It is demonstrated theoretically [136] that regions with different electrostatic potentials in

graphene can have the same transmission probability as the transmittance of layered struc-

tures in optics illuminated with normally incident TE waves if the incidence angle of electrons

in graphene and the width of the gated region are appropriately chosen.This quantitative

analogy can be useful for designing optical structures that correspond to graphene-based

devices. Then there are proposals of electron waveguides created by electrostatic potential

or by real magnetic barriers [82, 138]. Here the control the of the quasiparticle flux in a

graphene-based waveguide was theoretically investigated. Guided modes occuring in the

negative-zero-positive index metamaterial waveguide [141] and in symmetric velocity bar-

rier [140], have also been investigated. It is shown that magnetically induced waveguide

67

in graphene leads to strong confinement of Dirac fermions, regardless of its edge termina-

tions [142].The properties of unidirectional snake states have also been investigated in such

waveguides [139]. It was shown that for certain magnetic field profile, two spatially separated

counterpropagating snake states are formed, leading to conductance quantization which is

insensitive to backscattering by impurities or irregularities of the magnetic field.

It may be mentioned there are many other interesting articles in the relevant field and

because of space restriction it is almost impossible to refer to all of them which we regret. We

would also like to direct the readers some excellent reviews on electron transport in graphene

where some of the related issues was discussed in somewhat different context. In ref. [144]

the transport properties of graphene that includes theoretical and experimental work was

reviewed. In ref. [145] the charge and spin transport in mesoscopic graphene structures was

reviewd in detail. In ref. [146] the density and temperature-dependent charge transport in

doped or gated graphene devices was reviewed. In ref. [23, 24] the Klein tunnelling through

single and multiple potential barriers was reviewed. Properties of graphene in presence of

strong homegenous magnetic field was reviewed in ref. [147]. Electronic properties of bilayer

graphene was reviewed in detail in ref. [148].

To summarize, We have reviewed several aspects of charge carrier transport in graphene

through inhomogenous magnetic fields viz. magnetic barriers in detail. We reviewed ex-

perimental progress in designing such magnetic barriers at various length scales. Such bar-

riers suppress Klein tunnelling, thereby achieving confinement in graphene, which can be

seen through strong supression of transmission of electrons. We have shown that transport

through MVP barriers can be understood in terms of propagation of light through periodic

stratified media. This analogy can partially be attributed to the fact that equation describ-

ing Dirac like charge carriers in graphene and the Maxwell equations [143] are both linear

wave equations. We show that such transport properties through a singular magnetic barrier

can be much better controlled by the additional application of an electrostatic voltage. A

detailed optical analogy for transport through single and multiple EMVP barriers arranged

periodically was obtained highlighting optical analogues of phenomena such as TIR for pos-

itive and negative refraction and a Quantum Goos-Hanchen shift. One significant feature of

such transport in this regime of the simultaneous application of highly inhomogenous mag-

netic fields along with electrostatic potentials, is highly anisotropic and strongly dependent

on the sign of the voltage indicating possible device applications. We discuss collimation,

68

emergence of extra dirac points in this context. We also reviewed transport through such

magnetic barrier through unbiased graphene bilayer, describe reversal of Klein tunnelling,

anticloaking, common path interference etc. in such systems. The wide range of optical

analogues to charge transport in graphene may lead to practical devices such as Bragg re-

flectors, resonantors, waveguides etc. We hope this detailed review will augment theoretical

and experimental research in this field.

IX. ACKNOWLEDGEMENTS

We have benefited from discussions with many researchers including our students working

on graphene in the last few years at various places. We particularly thank G. Bhaskaran, F.

M. Peeters, R. Masir, M. Barbier, L. Levitov, R. Nandkishore, Leonardo Campos, Andrea

Young, P. Michetti, V. Fal’ko, E. McCann, Mandar Deshmukh, K. Sengupta, Subhasis

Ghosh, Sameer Grover, Sanjay Gupta, Nupur Gupta, V. Thareja for such discussions. We

thank A. Nogaret and IOP Publishing for giving us the kind permission to use one the

experimental figures published in ref. [17]. This work is supported by grant SR/S2/CMP-

0024/2009 from the Science and Engineering Research Council, DST, Government of India.

NA is supported by a CSIR fellowship.

[1] S. Datta, Electronic Transport in Mesoscopic Systems, Chapter 6 and 7, Cambridge Uni-

versity Press, (2005).

[2] W. van Haeringen, D. Lenstra Analogies in Optics and Micro Electronics, Dordrecht: Kluwer,

(1990).

[3] P. Yeh, A. Yariv and C.S. Hong, J. Opt. Soc. Am., 67, 423 (1977) : A. Yariv and P. Yeh,

Optical waves in crystals, Wiley Interscience, (2003).

[4] C. Kittel, Introduction to Solid State Physics, 7th Ed., John Wiley and Sons Pte. Ltd.,

Singapore (1996).

[5] U. Sivan, M. Heiblum, C. P. Umbach and H. Shtrikman, Phys. Rev. B 41, 7937 (1990).

[6] J. Spector, H. L. Stormer, K. W. Baldwin, L. N. Pfeiffer and K. W. West, Appl. Phys. Lett.

56, 1290 (1990).

69

[7] P. Wallace Phys.Rev. 71, 622 (1947).

[8] K. S. Novoselov et al., Science, 306, 666, (2004).

[9] K. S. Novoselov et al., Nature, 438, 197 (2005).

[10] Y. Zhang et al., Nature, 438, 201 (2005).

[11] V. V. Cheianov, V. Falko, B. L. Altshuler, Science 315, 1252 (2007).

[12] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).

[13] J. B. Pendry, Phys. Rev. Lett.,85, 3966 (2000).

[14] K. S. Novoselov et al., Nat. Phys., 2, 620 (2006).

[15] A. F. Young and P. Kim, Ann. Rev. Cond. Matt. Phys. 2 1-20 (2010).

[16] A. De Martino, L. Dell’Anna and R. Egger, Phys. Rev. Lett, 98, 066802 (2007).

[17] A. Nogaret, J. Phys. Cond. Matt. 22, 253201 (2010).

[18] J.Wrobel et. al., Phys. Rev. Lett. 93, 246601 (2004 ).

[19] S. Ghosh and M. Sharma, J. Phys. Cond. Matt. 21, 292204 (2009).

[20] A. Matulis, F. M. Peeters and P. Vasilopoulos, Phys. Rev. Lett, 72, 1518 (1994).

[21] M. Sharma and S. Ghosh, J. Phys. Cond. Matt. 23, 055501 (2011).

[22] A. H. Castro Neto et al, Rev. Mod. Phys., 81, 109 (2009).

[23] J. M. Pereira Jr et. al., Semicond. Sci. Technol. 25, 033002 (2010).

[24] P. E. Allain and J. N. Fuchs Eur. Phys. J. B 83, 301 (2011)

[25] R. Saito, G. Dresselhaus, and M.S. Dresselhaus Physical Properties of Carbon Nanotubes,

Imperial College Press 1998.

[26] M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 51 (2002) 1; R. C. Tatar and S. Rabii

Phys. Rev. B 25, 4126 (1982) ; J. C. Charlier, X. Gonze, and J.-P. Michenaud Phys. Rev.

B 43, 4579 (1991).

[27] J. C. Slonczewski, P. R. Weiss, Phys. Rev. 109, 272 (1958).

[28] G. W. Semenoff, Phys. Rev. Lett. 53, 2449 - 2452 (1984) .

[29] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015-2018 (1988).

[30] A. Geim and K. S. Novoselov, Nat. Mater., 6, 183 (2007)

[31] C. W. J. Beenakker, Rev. Mod. Phys., 80, 1337 (2008).

[32] J. Cserti et. al., Phys. Rev. Lett. 99, 246801 (2007).

[33] T. Ando, T. Nakanishi, J. Phys. Soc. Jpn. 67, 1704 (1998).

[34] A. F. Young and P. Kim, Nat. Phys. 5, 222, (2009)

70

[35] N. Stander, B. Huard, and D. Goldhaber-Gordon, Phys. Rev. Lett. 102, 026807 (2009).

[36] G. A. Steele, G. Gotz and L. P. Kouwenhoven, Nat. Nano. 4, 363 (2009)

[37] W. Greiner, Relativistic Quantum Mechanics, 3rd Ed., Springer (1997).

[38] T. Tudorovskiy et al, Phys. Scr. 146T, 014010 (2012).

[39] V. V. Cheianov and V. Falko, Phys. Rev. B, 74, 041403 (2006).

[40] E. B. Sonin, Phys. Rev. B 79, 195438 (2009)

[41] A. V. Shytov , M. S. Rudner and L. S. Levitov Phys. Rev. Lett. 101 156804 (2008).

[42] S. Sutar, E. S. Comfort, J. Liu, T. Taniguchi, K. Watanabe, and J. U. Lee, Nano Lett. 12,

4460, (2012).

[43] R. N. Sajjad, S. Sutar, J. U. Lee and A. W. Ghosh, Phys. Rev. B 86, 155412 (2012).

[44] N. M. R. Peres, A. H. Castro Neto, and F. Guinea, Phys. Rev. B 73, 241403(R) (2006); P.

G. Silvestrov and K. B. Efetov, Phys. Rev. Lett. 98, 016802 (2007).

[45] B. Trauzettel and Denis V. Bulaev, D. Loss, and G. Burkard, Nat. Phys. 3, 192 (2007).

[46] J. Milton Pereira, Jr., F. M. Peeters, and P. Vasilopoulos, Appl. Phys. Lett. 90, 132122

(2007).

[47] J. Milton Pereira, Jr., P. Vasilopoulos, and F. M. Peeters, Nano Lett. 7, 946 (2007).

[48] A. De Martino, L. Dell’Anna, and R. Egger, Solid State Commun. 144, 547 (2007).

[49] S. Khizroev and D. Litvinov J. App. Phys. 95, 4521 (2004); M. Lu, L. Zhang, Y. Jin, and

X. Yan, Eur. Phys. J. B 27, 565-70 (2002).

[50] M. Albrecht et al. IEEE Trans. on Mag. 99, 2323 (2003).

[51] A. Masseboeuf et al. Nano Lett., 9 (8), pp. 2803-2806 (2009).

[52] W.A. Challener et al. Nat. Photon. 3, 220 (2009).

[53] F. B. Mancoff, R. M. Clarke, C. M. Marcus, S. C. Zhang, K. Campman, and A. C. Gossard,

Phys. Rev B 51, 13269 (1995); F. B. Mancoff, L. J. Zelinski, C. M. Marcus, K. Campman,

and A. C. Gossard, ibid. 53, R7599 (1996); A. W. Rushforth, B.L. Gallagher, P. C. Main, A.

C. Neumann, C. H. Marrows, I. Zoller, M. A. Howson, B. J. Hickey, and M. Henini, Physica

E 6, 751 (2000).

[54] S.J. Bending, K. von Kiitzing, and K. Ploog, Phys. Rev. Lett. 85, 1060 (1990).

[55] A.K.Geim Pis’ma Zh. Eksp. Teor. Fiz. 50, 359 (1989) [JETP Lett. 50, 389 (1990)];

[56] M. L. Leadbeater et al., Phys. Rev. B 52, R8629 (1995).

[57] G. M. Gusev et al., Phys. Rev. B 53, 13641 (1996).

71

[58] M. Grayson et al., Physica E 22, 181 (2004).

[59] F. G. Monzon, M. Johnson, and M. L. Roukes, Appl. Phys. Lett. 71, 3087 (1997).

[60] M. Johnson, B. R. Bennet, M. J. Yang, M. M. Miller, and B. V. Shangbrook, Appl. Phys.

Lett. 71, 974 (1997).

[61] V. Kubrak, A. C. Neumann, B. L. Gallagher, P. C. Main, M. Henini, C. H. Marrows, and

M. A. Howson, Physica E 6, 755 (2000).

[62] S. J. Lee, S. Souma, G. Ihm, and K. J. Chang, Phys. Rep. 394, 1 (2004).

[63] F. M. Peeters and A. Matulis, Phys. Rev. B 48, 15166 (1993).

[64] I. S. Ibrahim and F. M. Peeters, Am. J. Phys. 63, 171 (1995).

[65] M. Johnson, B. R. Bennett, M. J. Yang, M. M. Miller, and B.V. Shanabrook, Appl. Phys.

Lett. 71, 974 (1997); V. Kubrack et al., J. Appl. Phys. 87, 5986 (2000).

[66] H. A. Carmona et al., Phys. Rev. Lett. 74, 3009 (1995); P. D. Ye et al., ibid. 74, 3013 (1995).

[67] A. Nogaret, S. J. Bending, and M. Henini, Phys. Rev. Lett. 84, 2231 (2000).

[68] K. S. Novoselov, A. K. Geim, S.V. Dubonos, Y. G. Cornelissens, F. M. Peeters, and J. C.

Maan, Phys. Rev. B 65, 233312 (2002)

[69] N.H. Shon and T. Ando, J. Phys. Soc. Jpn., 67, 2421 (1998).

[70] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Product (New York:

Academic) (1980).

[71] J. K. You, L. Zhang L and P. K. Ghosh, 1995 Phys. Rev. B, 52, 17243 (1995); A. Majumder

Phys. Rev. B 54, 11911 (1996); J. Reijniers J, F. M. Peeters, and A. Matulis, Phys. Rev. B

64, 245314 (2001); J. Reijniers and F. M. Peeters 2001 Phys. Rev. B 63, 165317 (2001);G.

Papp, P. Vasilopoulos and F. M. Peeters, Phys. Rev. B 72, 115315 (2005).

[72] N. Agrawal(Garg), S. Ghosh and M. Sharma, Proceeding of Indian national Science Academy,

77, 2 (2011).

[73] M. R. Masir et al. Phys. Rev. B. 77, 235443 (2008).

[74] M. Krawczyk and H. Puszkarski, Phys. Rev. B 77, 054437 (2008); S.D. Bader, Rev. Mod.

Phys., 78, 1 (2006).

[75] S.H. Sun et al, Science 287, 1989 (2000).

[76] H. Zheng et al, Appl. Phys. Lett. 80, 2583 (2002).

[77] K. von Bergmann et al, Phys. Rev. Lett. 96, 167203 (2006).

[78] J. Martin et al., 2008, Nature Phys. 4, 144 (2008).

72

[79] J. R. Williams et al, Science, 317, 5838, pp 638-641 (2007).

[80] A. K. Bhatt et. al., Solid State Commun., 152, 545-548 (2012).

[81] S. Pisana, P. Braganca, E. Marinero and B. Gurney, Nano Lett. 10, 341 (2010).

[82] Y.P. Bliokh, V. Freilikher and F. Nori, Phys. Rev B 81, 075410 (2010)

[83] S. Bhattacharjee, M. Maiti and K. Sengupta, Phys. Rev. B 76, 184514 (2007).

[84] N. Agrawal, S. Ghosh and Manish Sharma, arXiv:1206.0249 [cond-mat.dis-nn]

[85] L. Brey and H. A. Fertig, Phys. Rev. Lett. 103 046809, (2009).

[86] C. H. Park, L. Yang, Y. W. Son, M. L. Cohen and S. G. Louie, Phys. Rev. Lett. 101, 126804

(2008).

[87] C H Park, Y. W. Soon, L. Yang, M. L. Cohen and S. G. Louie, Phys. Rev. Lett.103, 046808

(2009).

[88] S. Grover, S. Ghosh and M. Sharma, Modelling Simul. Mater. Sci. Eng. 20, 045010 (2012).

[89] E. Milpas, M. Torres and G. Murguia, J. Phys. Cond. Matt. 23 245304 (2011).

[90] T. K. Ghosh et al., Phys. Rev. B 77, 081404(R) (2008).

[91] C.-H. Park, Y.-W. Son, L. Yang, M. L. Cohen, and S. G. Louie, Nano Lett. 8, 2920 (2008).

[92] Masir. M. Ramezani, P. Vasilopoulos and F. M. Peeters, Phys. Rev. B 82 115417 (2010).

[93] F. Goos and H. Hanchen, Ann. Phys. (Lpz.) 436 333-346 (1947).

[94] de Fornel F 2001 Evanescent Waves: From Newtonian Optics to Atomic Optics (New York:

Springer)

[95] P. R. Berman, Phys. Rev. E 66, 067603 (2002).

[96] I. V. Shadridov , A. A. Sukhorukov and Y. S. Kivshar Phys. Rev. Lett. 95, 193903 (2005)

[97] X. Chen, C-F. Li and Y. Ban, Phys. Lett. A 354, 161 (2006); X. Chen X, Y. Ban Y and

C-F. Li J. Appl. Phys. 105, 093710 (2009).

[98] X. Chen, C-F. Li and Y. Ban Phys. Rev. B 77, 073307 (2008).

[99] C. W. J. Beenakker, R. A. Sepkhanov and A. R. Akhmerov and J. Tworzydlo, Phys. Rev.

Lett. 102, 146804 (2009).

[100] D. Bohm, Quantum Theory, Prentice-Hall (New York, 1951), pp. 257261

[101] L. L. Chang, L. Esaki, and R. Tsu, Appl. Phys. Lett. 24, 593 (1974).

[102] M. Barbier, P. Vasilopoulos, F. M. Peeters, and J. M. Pereira, Phys. Rev. B, 79, 155402

(2009).

[103] L. Dell’Anna and A. D. Martino, Phys. Rev. B 79, 045420 (2009); See also the erratum at

73

Phys. Rev. B 80, 089901(E) (2009).

[104] L. Sun, C. Fang, and Y. Guo, J. Appl. Phys. 108, 063715 (2010); L. Sun, C. Fang, Y. Song

and Y. Guo, J. Phys.: Cond. Matt. 22 445303 (2010).

[105] A. K. Ghatak, K. Thyagarajan and M. R. Shenoy, IEEE J. Quant. Electron. 24, 1524 (1988).

[106] B. H. J. Mckeller amd G. J. Stephenson, Jr., Phys. Rev. C 35, 2262 (1987).

[107] B. Mendez, F. D. Adame and E. Macia, J. Phys. A: Math. Gen. 26, 171 (1993).

[108] P. Strange, Relativistic Quantum Mechanics, Cambridge University Press, (1998).

[109] M. Barbier, F. M. Peeters, P. Vasilopoulos and J. M. Pereira, Jr., Phys. Rev. B 77, 115446

(2008).

[110] A. Krakovsky, Phys. Rev. B, 53, 8469, (1996).

[111] R. Naseer, K. Sabeeh and M. Tahir, Phys. Rev. B 81, 085402 (2010).

[112] C. H. Park et al, Nat. Phys., 4 3, 213-217 (2008)

[113] Y. P. Bliokh et al, Phys. Rev. B, 79, 075123. (2009)

[114] L. D. Anna and A. De. Martino, Phys. Rev. 83, 155449 (2011).

[115] V. Q. Le et al., J. Phys. Condens. Matt. 24, 345502 (2012).

[116] Vinu Lukose, R. Shankar, and G. Baskaran, Phys. Rev. Lett. 98, 116802 (2007).

[117] L. Z. Tan, C. H. Park and S. G. Louie, Phys. Rev. B 81, 195426 (2010).

[118] L. D. Landau and E. M. Lifshitz, Classical theory of fields, 3rd Ed., Pergamon, London

(1971).

[119] N. Gu et. al., Phys. Rev. Lett. 106, 066601 (2011).

[120] E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96 086805 (2006) ; E. McCann, Phys. Rev. B

74 161403(R) (2006).

[121] S. Park and H.-S. Sim, Phys. Rev. B 84, 235432 (2011)

[122] C. J. Poole, Solid State Commun. 150, 632 (2010).

[123] Masir. M. Ramezani, P. Vasilopoulos and F. M. Peeters, Phys. Rev. B 79, 035409 (2009);

Masir. M. Ramezani et al, Appl. Phys. Lett. 93, 242103 (2008).

[124] J. Nilsson et. al., Phys. Rev. B 76, 165416 (2007); M. Barbier, P. Vasilopoulos, F. M. Peeters

and J. M. Pereira, Phys. Rev. B 79, 155402 (2009).

[125] B. D. Terris and T Thomson, J. Phys. D: Appl. Phys. 38 R199 (2005).

[126] N. Agrawal (Garg), S. Grover, S. Ghosh and M. Sharma, J. Phys. Cond. Matt. 24 175003

(2012).

74

[127] G. Breit and E. Wigner, Phys. Rev. 49, 519 (1936).

[128] N. Gu, M. Rudner and L. Levitov Phys. Rev. Lett. 107, 156603 (2011).

[129] H. Min and A. H. MacDonald, Phys. Rev. B 77 155416 (2008); H. Min and A. H. MacDonald,

Prog. Theor. Phys. Suppl. 176 227 (2008).

[130] M. Killi et. al., Phys. Rev. Lett. 107, 086801 (2011); M. Barbier, P. Vasilopoulos, and F.

M. Peeters, Phys. Rev. B 81, 075438 (2010); M. Barbier, P. Vasilopoulos, and F. M. Peeters,

Phil. Trans. R. Soc. A 368, 5499 (2010).

[131] R. Nandkishore and L. Levitov, Proc. Natl. Acad. Sci. 108, 14021 (2011).

[132] P. San-Jose, E. Prada, E. McCann and H. Schomerus, Phys. Rev. Lett. 102, 247204 (2009)

[133] P. Michetti et al, Nano Lett. 10, 4463 (2010)

[134] L. C. Campos et. al., Nature Commun., 3, 1239 (2012).

[135] D. Dragoman, J. Opt. Soc. Am., B 27, 1325 (2010).

[136] I. Mihalache and D. Dragoman, J. Opt. Soc. Am., B, 28, 7, pg. 1746 (2011).

[137] D. Dragoman, J. Opt. Soc. Am., B, 29, 6, pg. 1528 (2012).

[138] C. E. P. Villegas and M. R. S. Tavares, App. Phys. Letts., 101, 163104 (2012).

[139] T. K. Ghosh, Phys. Rev. B 77, 081404(R) (2008).

[140] J-H. Yuan et. al., J. App. Phys., 110, 103706 (2011).

[141] M. Shen et. al., Opt Exp. 18, 12779 (2010).

[142] N. Myoung et. al., Phys. Rev. B 83, 113407 (2011).

[143] D.W. Berreman, J. Opt. Soc. Am., 62, 502 (1972).

[144] N. M. R. Peres, Rev. Mod. Phys. 82, 2673 (2010); N. M. R. Peres, J. Phys. Cond. Matt. 21

323201 (2009).

[145] A.V. Rozhkov et. al., Physics Reports, 503, 77114 (2011).

[146] S. D. Sarma et. al., Rev. Mod. Phys. 83, 407 (2011).

[147] M. O. Goerbig Rev. Mod. Phys. 83, 1193 (2011).

[148] E. McCann and M. Koshino, arXiv:1205.6953 [cond-mat.mes-hall].

[149] V. N. Kotov et. al., Rev. Mod. Phys. 84, 1067 (2012).

[150] F. Molitor et. al., J. Phys. Cond. Matt. 23 243201 (2011).

[151] F. Bonaccorso, Z. Sun, T. Hasan and A. C. Ferrari, Nature Photonics 4, 611 - 622 (2010).

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